When and how did the term "Nazi" start replacing the term "Germans" in the context of WW2?

When? How? As soon as someone wanted to create sympathy for the Germans, despite all the atrocities committed by willing executioners in an extermination war for global dominance. This kind of sympathy is a hard sell once you know how the war was fought, how many people were mistreated, tortured, killed and what a fate the Germans had in store for those that would survive the war under their rule. Also as soon as the term "nazi" became a selling point for poor history in magazines, movies and TV. And almost germane, as soon as this is simply a misguided to desire to delineate a specific timeframe of German history. The last point unfortunately skids into pseudo-history as it obscures agency and responsibility. Popular sloppiness, psychological hygiene and political opportunism go hand in hand when this common formula is encountered.

Who is or was "a Nazi"?
A member of the NSDAP? Someone who embraces Nazism? Someone who does Nazi-things or thinks Nazi-Thoughts? A racist, an antisemite, a war criminal?

Who is or was "a German"?
A citizen of the German Reich in 1918, 1924, 1933, 1937, 1939, 1945? Some former German citizens in 1918 became Danes, Poles, French and Belgians. Some Austrians or Czechs became German citizens before 1939. Is it a Wolgadeutscher, a Banater, a Siebenbürger Sachse, someone from South Tyrol? (People called Volksdeutsche, the nazi-term for ethnic Germans)?

Is there more of a difference or more of a congruence if not identity for the timeframe in question?

This is no linear process and the chancellor of West Germany, Adenauer, certainly saw the continuity of identity, for him the Germans were the Nazis and a distinction quite non-sensical:

When Chancellor Konrad Adenauer was asked in early 1955 if there should be an official event marking the 10th anniversary of the liberation from the Nazis he answered, tellingly: "You don't celebrate your defeats."

Although these thoughts alone make the issue a bit more complicated than commonly taught in school, it is nevertheless a pondering quest to differentiate between these terms, even if a general definition of them might not fit in all cases to be discussed.

Up until 1945 almost all Germans just were Nazis. At the end of the war the NSDAP had 8.5 million members. The Wehrmacht sent 18,200,000 men into battle and war crimes, while the SS only had "Employees 800,000 (c. 1944)". From 1936, at the latest, until start of the war the popularity of the government went up, approval ratings improved and were only for a short time on hold since skepticism about starting the war gained some ground. That was put to rest relatively quickly and Hitler was at the all time high of popularity and approval when France capitulated. Two things made this thunderous rise in approval come down again: loosing the war against the Soviet Union, symbolised by Stalingrad and declaring war on America. But only very slowly so. So in retrospect, only up until 1942 almost all Germans were Nazis by heart.
That is not only a useful story to tell your people at home and an army on its way to invade the enemy country.

Hitler Endorsed by 9 to 1 in Poll on his Dictatorship, but Opposition Is Doubled

Examples for equating Germany with Naziland are countless:

Never in the field of human conflict was so much owed by so many to so few. All hearts go out to the fighter pilots, whose brilliant actions we see with our own eyes day after day; but we must never forget that all the time, night after night, month after month, our bomber squadrons travel far into Germany, find their targets in the darkness by the highest navigational skill, aim their attacks, often under the heaviest fire, often with serious loss, with deliberate careful discrimination, and inflict shattering blows upon the whole of the technical and war-making structure of the Nazi power.
Churchill's "The Few"-speech

https://youtu.be/FRRxG4L4upA?t=354

The most defining aspect of Nazi government in recent memory was the industrialised extermination of Jews, Roma and other whole groups of people. While this crime of unspeakable extent is considered important today, neither this crime nor any other war crime was seen as important by the vast majority of Germans in the day. This is not to say there were some non-racists, some resistance members and so on. But is absolutely accurate to describe the inhabitants of Nazi-Germany as Nazi-Germans. The Germans were Nazis. The crimes were committed by Germans (and to a much smaller extent, by Poles, by Ukrainians etc). Not every member of the SS, the Reichsbahn and not even the upper echelons of Wehrmacht was also in the NSDAP. The crimes were not carried out by party-members exclusively and the war was not carried out by party-members. After the war especially the surviving soldiers tried to made a point of having been in the "saubere Wehrmacht" ("the clean Army"), despite carrying out the war, despite all the war crimes, and despite things like Kommissarbefehl and anti partisan guerilla and mass shootings of Jews and so on.

That has some consequences today. If you correctly say the Germans started the war, fought it as they did and committed war crimes, built the concentration camps and extermination camps, then this carries over into judging not the actual perpetrators, most of them now also dead but their sons and grand-daughters and every other now living German. That goes not only occasionally a bit too far for historical accuracy:

Opa war kein Nazi – Grandpa wasn’t a Nazi – the research found that the younger generation felt a need to separate their beloved grandparents from the dark past, to dissociate them from the bad Nazis they had heard about.

If you are a now amiable French a skeptical British^* in a shared European Market and Union, allies in the same military pact etc. that is quite an uncomfortable thought. Together in one boat with those beasts of un-human cruelty? It's much better to alienate the description to some Nazis and think of the Germans being just like the Austrians the first victims of Nazism.

This started at the time, but not because, of the Nuremberg trials to get a wide-spread hold in common parlance. Before these were concluded the Cold War had started and both sides of that war needed new allies that they found in the Nazis Germans, East and West.

Very soon after the program started, due to the emergence of the Cold War, the western powers and the United States in particular began to lose interest in the program, and it was carried out in an increasingly lenient and lukewarm way until being officially abolished in 1951.

Unofficially the strictest prohibitions against fraternisation were already largely ignored and some previous plans to condemn the whole nation were abandoned.

Finally, a fourth premise in occupation plans stemmed from conclusions about the German mentality and the expectation that the Germans would attempt to trick or corrupt Allied officers by spreading pro-Nazi, pro-German, militarist propaganda. Perhaps the most important result was the ‘non-fraternization’ rule, which prohibited Allied troops stationed in Germany from maintaining any personal contacts with the German population. By early 1944, instructions informed troops not just about the ‘mission of the occupying forces’, but also about the ‘characteristics of the German people, their probable attitude towards the forces of occupation and the type of propaganda which they are liable to employ’. _{From "A Hard Peace? Allied Preparations for the Occupation of Germany, 1943–1945"}

The allied programme of denazification, previously quite strict, came to an end and especially in the West most of those not already on special lists or already convicted were allowed to regain their positions. Among them teachers, judges, civil servants, police and politicians. Before the Cold War got into swing the terms Nazi and German were synonyms, and rightly so. For the Germans themselves this process started of course immediately, that is at the start of May 1945 or even before, whenever their town or village was "liberated from the Nazis".

Examples for these changing attitudes on the side of the allies can be shown quite easily if you compare the Morgenthau plan and public opinion at the time the allies reached the concentration camps with the American protection for the careers of SS-Sturmbannführer Wernher von Braun and Reinhard Gehlen.

Despite viewing 'Nazis' as a finished business threat whose position in most minds of the Western world was now taken over by communism, there were also some key points in separating Nazis and Germany, that is slowly growing sympathy for Germans/Germany now slowly seen as "also victims":

• atrocities and brutality being committed by Eastern powers in expelling millions of Germans from the East,

'A Tragedy on a Prodigious Scale': Churchill later criticized the brutal behavior of the Poles and Soviets, calling it "a tragedy on a prodigious scale" -- as if ethnic cleansings had ever been anything but tragic.

• Hunger winters before the establishment of two Germanies,

• the Berlin Blockade

• and finally German admission into Nato ("A major reason for Germany's entry into the alliance was that without German manpower, it would have been impossible to field enough conventional forces to resist a Soviet invasion.[")

• as well as forerunner organisations of the European Union.

He shows how American policy vacillated from the righteous rhetoric of Robert H. Jackson to the ultimate embrace and rearming of West Germany, all because of Cold War fears that dominated American foreign policy from 1948 to 1958. Indeed, he implies, little is left of the IMT war crimes Nuremberg trials but nostalgia. By the late 1950s the Allies released their last convicted war criminals. A High Commissioner to Germany such as John J. McCloy--another Wall-Street-trained attorney in the tradition of Elihu Root and Robert Lansing--well reflected the new Cold War mentality.
_{(From a Review by Jonathan Lurie (Rutgers University, Newark; Peter Maguire. "Law and War: An American Story." New York: Columbia University Press, 2000.)}

John Mearsheimer: "Instability in Europe After the Cold War" Second, the horror of Germany's murderous conduct during World War II should be distinguished from the scope of the aggressiveness of German foreign policy.38 Germany was indeed aggressive, but not unprecedentedly so. Other states have aspired to hegemony in Europe, and sparked wars by their efforts; Germany was merely the latest to attempt to convert dominant into hegemonic power. What was unique about Germany's conduct was its policy of mass murder toward many of the peoples of Europe. The causes of this murderous policy should not be conflated with the causes of the two world wars. The policy of murder arose chiefly from domestic sources; the wars arose mainly from aspects of the distribution and character of power in Europe.

This psychological aversion and political opportunism is even more pronounced from a self-perspective in Germany, let alone Austria. Austria had its own fascist government before Hitler Germany annexed Austria into the Reich and "allowed" Austrians into the NSDAP, the SS the Wehrmacht. Many of them wholly enthusiastic about the things to come. After the war was lost the Austrians were quick to distance themselves from anything, and that quite successfully. In Germany even former party members started to use the phrase "and then Hitler came to power" meaning some otherworldly creature and his minions took control, not only control over political affairs but asserting mind control over an unwilling and powerless population. That is demonstrably just not true.

According to Lacau, the defeat of the workers’ parties at the hands of fascism was connected to these parties limiting themselves to proletarian class-discourses, while the Nazis developed a populism that was able to occupy the contradictions between the ruling power-bloc and the ‘people’, and incorporate them into a racist anti-democratic discourse.[…]
Pointing out the omnipresent violence in Nazism is obviously insufficient in explaining how the German fascists successfully mobilised the masses until the end and unleashed an enormous potential of youth ‘idealism’ for their purposes.[…]
Taking up these (and other) attempts to grasp the ideological appeal of fascism, the PIT investigated how the Nazis understood in an unprecedented manner ‘how to organise self-alienation as enthusiastic self-activity’.
_{(From Jan Rehmann: "Theories of Ideology The Powers of Alienation and subjection", Brill: Boston, Leiden, 2013. Cited here to show the concept again and to demonstrate that it worked both ways, before, during and after the war.)}

This process of alienating agency is in a sense justified. Insofar that relatively young Germans of today may still enjoy the profits from the effects of the war, but they did not commit the atrocities themselves. They may be called innocent. ("Profiteering" is in the sense that West Germany was after the war more poweful than before and that not in the least because it benefited from being an untouched "Raubgemeinschaft" (Cf. Aly below))

This process of alienating agency is in a sense not justified, insofar that to a certain extent it is the same fallacy as using the phrase "war broke out". As if it was a natural thing, an unstoppable catastrophe, with no-one in charge, nobody causing it, without actors, without people making decisions and then none of the people carrying out those decisions, whether forced to do so or with varying degrees of approval.

It is about distancing one's own mind from the crimes and atrocities as far away as possible. Whether German or Austrian, American, French or English. Today, the word "Nazi" is much less a historical description but an insult. Just as it ever was for any real non-Nazi (and Adolf Hitler himself?). The NSDAP has ceased to exist, and there are Nazis and Neo-Nazis everywhere. In the context of the question here: "Nazi" did not replace "German" in general when the topic is Germany from 1933–1945. It has now often replaced the term "German" when the topic is touching agency and guilt. Either unintentionally benign or aiming at absolution on purpose.

So with either term you end up with some form of not completely just imprecision. Calling all Germans from that time Nazis penalises the small number of resisting Germans, Antifas and the victims of persecutions, but it gives the right notion that the vast majority supported the war, the persecutions and doubled their guilt after the war when they insisted on personal innocence, clean conscience and "we didn't know" (not "…any better" but really a very wilful ignorance for everything that was plain to see.) –– Calling the most powerful axis member "the Nazis" underplays the fact that almost all Nazis were Aryan Germans and that the majority of all Germans went with flying colours and the greatest approval into the abyss of criminal cruelty and crimes against humanity.

Kershaw notes that Hitler's popularity within Germany – and German support for the war – reached its peak when he returned to Berlin on 6 July from his tour of Paris.

Swiss psychoanalyst Carl Jung wrote an influential essay in 1945 about this concept as a psychological phenomenon, in which he asserted that the German people felt a collective guilt (Kollektivschuld) for the atrocities committed by their fellow countrymen, and so introduced the term into German intellectual discourse. Jung said collective guilt was "for psychologists a fact, and it will be one of the most important tasks of therapy to bring the Germans to recognize this guilt." After the war, the British and US occupation forces promoted shame and guilt with a publicity campaign, which included posters depicting concentration camps with slogans such as "These Atrocities: Your Fault!" (Diese Schandtaten: Eure Schuld!).
(From German collective guilt, but compare Deutsche Reaktionen)

These thoughts and observations apply to some explicit and implicit, conscious und unconscious types of usages. It does not represent a rule to be applied generally and in every case. It does however portray some general trends. Sometimes calling the war-faring Germans of that time "the Nazis" might be a fitting description, sometimes it is used or has the effect to blur the lines of what really happened. Whether this is unintentional or deliberate is to be judged on a case by case basis. Regarding textbooks on the subject it seems fair to assume in most instances that quite often a certain well intentioned pedagogy will use "the Nazis" in order not stir up controversy that might flare up in young minds when they start to think about current Germans.

Conclusion:
When?
Depending on political leaning and and your own position: in West-Germany it started in every village as soon as it was taken, solidified during the phase of Westintegration, and despite some left-wing outliers it became firmly established view after the liberal turn and the Historikerstreit. Other members of "the West" followed a similar pattern, starting soon after tensions with the Soviet Union became apparent and increasingly so as time went by, in parallel as present day Germany became a rehabilitated member of the United Nations or 'the international community'.

Why?
It started with practical power politics in that Germans wanted to distance themselves from the guilt and the former allies needed the economic might and manpower. Manpower is somewhat undermined in efficiency and willingness to cooperate if constantly reminded of indelible guilt. With every year passed since then it is somehow increasingly seen as unjust to apply collective guilt to a whole people since the individual perpetrators were dying out. Shoa survivors often disagree, but they are the among the last to still hold this more than justified grudge.

Answer needed despite not being asked explicitly:
Is it really always the case that Germans and Germany from 1933–1945 are now always called Nazis and Nazi-Germiny or the Nazis? –– No. "Nazi-Germany" is a relatively specific and accurate description of the state and nation from 1933–1945. But a trend is observable and some of the reasons – sometime utter banality some of them very complicated – for that trend were in need of being discussed here. Underlying motivations are politeness, precision on the positive side, transmogrification of reality and retrograde absolution on the other side.

If you read "the Nazis attacked Russia" instead of "the Germans attacked the Soviet-Union" you might see a a peculiar difference made with that distinction and you are quite right to infer something about the nack for or lack of historical accuracy and the motivations of the author of those lines.

Main sources:
Ian Kershaw: "The Nazi Dictatorship. Problems and Perspectives of Interpretation", Bloomsbury Academic: London, Oxford, 2015.
Götz Aly: "Hitlers Volksstaat. Raub, Rassenkrieg und nationaler Sozialismus", Fischer: Frankfurt aM, 2006.
Stephan Grigat: "Postnazismus revisited. Das Nachleben des Nationslasozialismus im 21. Jahrhundert", ça ira: Freiburg, 2012.
Robert G. Moeller: "War Stories. The Search for a Usable Past in the Federal Republic of Germany", University of California Press: Berkeley, Los Angeles, 2001. (Relatively accurate mainstream description of the German side of this debate in the chapter Epilogue, p 171–198.)

For the most part, after the war ended as a means of discriminating between the former enemy state and the new allied state.

During the war itself the common pejoratives in use by English speakers were:

• Hun - more common in WW1 than WW2, mostly by British soldiers.

• Fritz

• Heinie - favoured by Canadian and U.S. soldiers.

• Jerry - from a resemblance of their helmet to British chamber pots or Jereboams

• Kraut - from the German dish sauerkraut.

• Boche

as well as others less common.

There is a conflict between Public History and scholarly historiography here.

Public History in the West (“brainwashed boys”) and East (“Hitlerite fascists”) sought to create the myth of the Good German. This allowed imperialist states to claim that their Germans or their Germany was a normal state filled with normal people. And by contrast not the inheritor state of a general population that wilfully demanded aggressive war and genocide. “Nazi” or “Hitlerite” Germany is an exculpatory ideological claim.

The historiography differs, believing that all of the Germanys have been normal states filled with normal people: and that aggressive war and genocide are (sadly) normal actions by states and peoples. Where the historiography differs is over the issue of intentionalism versus functionalism. Intentionalists assign greater emphasis over policy to the central state apparatus and Nazi leadership. Functionalists view policy and enactment as a system of agents with multiple moments where local or popular power can erupt and create policy or action. The former are more likely to emphasise “Nazi Germany” the latter to emphasise the “Nazification of Germany” or “Germany’s broad agreement with Nazi goals and policy.” Intentionalism’s impact in the historiography has waned, particularly in the light of the influence of Marxist, structuralist, sociological, culturalist and institutional type analyses. Functionalist or functionalist influenced historiographies are dominant in the scholarly discourse.

Why does public history differ from the scholarly? Brainwashed boys, or Nazis in Black Leather are exculpatory myths. They project evil onto “the other.” They are circulated by cultural works where “brainwashed boys” versus “the righteous individual” make for good art. They are circulated by states which wish to make clear that they are not “Nazi” states and that therefore should not have their potential for aggressive war or genocide inquired into. Additionally militaria as a publishing field promotes the “myth of the clean Wehrmacht” so that writers and readers can enjoy their tanks without the blood of slave labourers in their welds and civilians in their tracks.

Scholarly history is written primarily to circulate ideas amongst a scholarly public and its periphery. It has difficulty spreading its messages, due to the small size of the scholarly community and periphery. The message regarding the general potential of modern states and peoples to warmongering and genocide is deeply uncomfortable and resisted. “Humanities popularisation” is even less well funded than science popularisation; and, the Galleries Libraries Archives and Museums sector is often under ideological control by the state (however resisted.). Popular media around history, most notably a US television channel, refuses scholarly super intention as unprofitable, and prior to this was known as “The Hitler Channel” due to its fixation.

Popular history is calming and well funded; but unjustifiable. Scholarly history is problematic and poorly funded; but accurately represents best readings of sources.

It started around Teheran Conference, after the knowledge of german war crimes in territories liberated by the Soviet Union. Since only war criminals should be put on trial, they started to make the distinction between criminals and the rest of the population. Later on criminals started to be associated to nazis.

By Yalta Conference, the declaration used the word nazi several times. So then it was clear that the concept nazi was prevalent as the cause of war.

Near the end of the war, it was required to make a strong distinction between germans and nazis, in order to prevent abuses from allied forces against german civilians and prisioners. And also to give an excuse to remove former nazis from civil goverment during the occupation of Germany.

I suspect this is more a cause of change than anything else.

When we speak today of activities involving Russia in the 20th century we speak of Soviet Russia, or Bolshevist Russia. Similarly one may speak of imperial England.

The word Nazi characterizes Germany of the last 1930's and 1940's because the party dominated the political landscape and many unique features were attributed to it (propaganda, unique economic policy, harsh racism and nationalism).

So how did it happen that we no longer refer to activities conducted by German State and German ordinary folks during the period of 1939-1945 by the adjective "Germans" but we use "Nazi" instead?

Reference to the German State of 1939-1945 is implicitly a reference to the Nazi party. So it is natural to use the word Nazi in order to make a justifiably clear distinction between that government and it's predecessors and successors.

However I don't think we no longer refer to Germans of that period as Germans.

Most people know that not all Germans of that era were ever members of the National Socialist (Nazi) party. Many Germans would have been members of other political parties or of none.

In particular, we do not, under any circumstances, refer to as Nazis those Germans who were members of persecuted minorities and many or most of whom were murdered in Nazi death camps.

Many products of that era are referred to as German products and not as Nazi products, even when strongly associated with the results of Nazi policy. For example:

The Messerschmitt Bf 109 is a German World War II fighter aircraft that was the backbone of the Luftwaffe's fighter force.

or

The V-2 [...] was developed during the Second World War in Germany as a "vengeance weapon",

Note that the author could have written "in Nazi Germany" or "by the Nazis" but did not.

or

Closely cooperating with the SS (especially on the Eastern Front), the German armed forces committed numerous war crimes and atrocities, despite later denials.[9] By the time the war ended in Europe in May 1945, the Wehrmacht had lost approximately 11,300,000 men

and

Oskar Schindler (28 April 1908 – 9 October 1974) was a German industrialist and a member of the Nazi Party

My emphasis.

So I don't think the premise is justified, people often use the adjective "German" to refer to people, organisations and products of that era.