When was the last moment that Dutch, German and Swiss German were the same language?

There will be no definite answer for this, as the available evidence does not provide a straight avenue for an answer with much certainty. Furthermore is the timeframe only very roughly to sketch, as the most basic distinction criteria present a patchwork and even one stretching over a long time period. Arguments can be made for well before the year 1 as well as for Early Modern Period, as well as even rejecting the whole concept as an anachronistic reconstruction based on mere hypotheticals.

A summary of perpetual difficulties present is found in — Hans Frede Nielsen: "The Germanic Languages. Origins and Early Dialectal Interrelations", The University of Alabama Press Tuscaloosa London, 1989.

An approach with much leeway might be to look simply for a common ancestor, which would be West Germanic in this case. But the problems already look like mountains at this stage:

We do not have much historical or linguistic evidence for this, as those people in question were not much into the habit of writing things down in meaningful amounts on not perished by now material.

As such the very existence of One West Germanic is called into question sometimes by even the majority of scholars (says Wikipedia: Existence of West Germanic proto-language):

Most scholars doubt that there was a Proto-West-Germanic proto-language common to the West Germanic languages and no others, but a few maintain that Proto-West-Germanic existed.

What we do have is our theoretical reconstruction based on linguistic analysis, which is weighed down with problems, and archaeological evidence, which is just a proxy and even more influenced by certain politics.

As such those early groupings of seen as distinct languages into a family tree are ripe with decisions and thus opinions. One variant looks like this:

The individual languages of the West GMC group are the next attested, Old English (OE) and Old High German (OHG) being documented from roughly the eighth century, and Old Saxon (OS) in the ninth century, Frisian (FR) and DU from still later periods.

— Wayne Harbert: "The Germanic Languages", Cambridge University Press: Cambridge, New York, 2007, p15.

We see that West Germanic seems to split, but is really just grouped into three different branches at the time between migration period and 800, but that North Sea Germanic did impact little the development of Dutch. The influences of both Low German and Frisian, both still minority languages in The Netherlands are not depicted, and the main heritage is from Low Franconian, while its main sister language High Franconian seems to merge back into Alpine Germanic Languages like the modern Standard German and Alemannic Swiss German dialects.

A summary for the West Germanic branch as really a big bunch of closely related dialects — but a bunch that in fact did exist — in close a productive contact would be:

Whether there was ever a more or less unitary Northwest Germanic language has been a matter of dispute. In my opinion the number of signiWcant innovations which North and West Germanic unarguably share, though admittedly small, is large enough to justify positing such a unity. By contrast, the innovations shared by East and North Germanic are extremely few and can have resulted from parallel development, while those supposedly shared by East Germanic and the more southerly dialects of West Germanic are actually shared retentions which prove nothing (cf. e.g. Krause 1968: 48–52). That North Germanic is itself a unitary subgroup is completely obvious, as all its dialects shared a long series of innovations, some of them very striking (see Noreen 1923 passim). That the same is true of West Germanic has been denied, but I will argue in vol. ii that all the West Germanic languages share several highly unusual innovations which virtually force us to posit a West Germanic clade. On the other hand, the internal subgrouping of both North Germanic and West Germanic is very messy, and it seems clear that each of those subfamilies diversiWed into a network of dialects which remained in contact for a considerable period of time (in some cases right up to the present).

— Don Ringe: "A Linguistic History of English. Volume I. From Proto-Indo-European to Proto-Germanic", Oxford University Press: Oxford, New York, 2006, p 213–214.

The messiness of this situation is amplified when we now analyse our own very concept of 'this is one language, and this is a different one'.

It is customary to divide Germanic into East Germanic, with Gothic as its prominent member, North Germanic, with Icelandic, Faroese, Norwegian, Danish and Swedish, and West Germanic (sometimes 'South Germanic'), with German, Yiddish, Pennsylvania German, Dutch, Afrikaans, Frisian and English. If we relate this variety to the one Common Germanic language of two thousand years ago, we face the question of how we got from the one parent language to the three branches and to the dozen or so descendant languages. One factor to bear in mind is that every language is inherently variable. A language only exists through speakers that speak an idiolect, and typically share a dialect - and sociolect - with the people with whom they communicate most often or want to be associated. Thus some degree of dialectal variation must have prevailed in Common Germanic too, an assumption plausible also on purely linguistic grounds. Standard methods of linguistic reconstruction sometimes lead to two reconstructed forms rather than only one, suggesting that Common Germanic allowed both. Thus the inherent linguistic variation within Common Germanic itself may safely be taken as a partial explanation of later diffusion, in particular, of the distinction between North and West Germanic.

A second factor responsible for the variety in Germanic is migration. When speakers move away from their homeland and cut or strongly diminish communication with those who stay behind, the inherent tendency for dialect variation increases. The migrants, moreover, may come into contact with speakers of another language, which may alienate either language, in varying degrees, from the language of the previous generations. It may also lead to the disappearance of one or even both of the languages. A distinction may be made in terms of the language variety with which the migrants left. Did they leave with Common Germanic, with a branch of Germanic like relatively undifferentiated North Germanic, or with a fully differentiated separate Germanic language like English? Germanic illustrates each of these types and scenarios. […]

As for West Germanic, tribal groups of Angles, Saxons and Jutes invaded England during the fifth and sixth centuries, and the Langobard(ic) (Lombard) tribe moved into Italy. Whereas the southward expansion proved unsuccessful (by the end of the first millenium Langobardic was basically extinct), the westward expansion led to modern English. […]

A third factor needed to explain how one ancestral language relates to a dozen descendant languages is standardization. Without this concept one would still not know why Swedish, Danish and Norwegian are considered different languages, even though mutual intelligibility is very high, whereas some northern and southern dialects of German, which are hardly mutually intelligible, are not considered separate languages.

Standardization is the process whereby a community, typically a literate one, imposes a uniformity on its language in response to a growing desire of political, religious or cultural authorities for improved communication across dialects. The standard which then emerges is typically based on dialects that are (a) spoken in the economically and culturally strongest region; (b) deemed 'authentic' in a way that satisfies a sense of national identity in search of a national language; and/ or (c) more highly cross-dialectally intelligible than others. Early catalysts were the printing press; […]

— König, Auwera

This 'standard makes a language' to be recognised as one, where much emphasis is put intentionally on distinction and group identity is partly visible if you cross the borders of Northern Germany and The Netherlands, with the local dialects still being different from the 'Official Language', but changing rather gradually. The differences in trying to standardise this language are very neatly illustrated in just reading the different Wikipedia page titles for the item: English: Low Saxon, Standard German Niedersächsisch and Niederdeutsche Sprache is in official Dutch: Nedersaksisch, but in 'Dutch Low Saxon': West Leegsaksisch and in 'German Low Saxon': Plattdüütsch.

How much ever dispute goes into this separation between languages as distinct analytical entities, whether 'it's clearly different' or 'more or less the same on a long spectrum, where the ends seem clearly different, but drawing the dividing line is a hard thing to do', giving us a quite wide timeframe:

The dialects of German subdivide into Low German (Niederdeutsch, Plattdeutsch) and High German (Hochdeutsch). The former are spoken in the north of Germany, the latter in the centre and the south. In linguistic terms, the criterion is the degree to which the dialects have been affected by the so-called 'High German Consonant Shift' : Low German has not been affected by it; German partially, and Southern or 'Upper' German (almost) completely. Modern Standard German developed primarily on the basis of the late medieval chancery language of the court of Saxony and the East Central dialect area around Dresden. In the course of the sixteenth and seventeenth centuries this written language gained gradual acceptance throughout the entire German-speaking area, in part because of the economic power of Saxony and the position of the dialect, intermediate between Low and Upper German and thus more widely comprehensible than either, and in part because Luther made it the language of the Reformation. In this process of geographical expansion, the emerging Modern German standard ousted - but was also influenced by - competing regional standards, the Low German standard of the Hanseatic League in the north, and the Upper German 'Common German' (gemeines Deutsch) in the south. The spoken standard spread much later and is based on the North German pronunciation of the written standard, bearing witness to the fact that by the end of the eighteenth century Saxony had lost political power and cultural prestige to Prussia. The expansion of the spoken standard was never completed, however: both in Switzerland and in Luxembourg the local dialects, when spoken, have the social prestige normally associated with a standard language.

High German is documented first in runic inscriptions and glosses, and later in clerical texts, a phase called Old High German (until c.1100), followed by Middle High German (until 1400 or 1500), the period of courtly and epic poetry, then Early New High German (until c.1650), which laid the foundations of the modern New High German (from c.1650). For Low German, one distinguishes between Old Low German or Old Saxon (until c. 1100), Middle Low German (until 1400 or 1500), contemporaneous with the heyday of the Hanseatic League, and thereafter New Low German.

— Ekkehard König & Johan van der Auwera: "The Germanic Languages", Routledge: London, New York, 1994. (esp Carol Henriksen & Johan van der Auwera: "The Germanic Languages" & Peter Eisenberg: "German", Dutch Georges De Schutter: "Dutch").

The German language, of which Swiss Standard German would be one branch as well as most of the Swiss German mainly Alemannic dialects, standardised quite late. In comparison, even 'the Dutch language' was for a long time not one language:

Although the geographical distribution of Dutch is rather limited, there is a wide variety of regional dialects, the mutual intelligibility of which is often low. The diversity may be traced back to at least two sets of factors, one intra-, one extralinguistic in nature.

First of all, the language developed in a geographical area in which no fewer than three or even four major dialects of continental West Germanic come together: Frisian, Saxon and Low Franconian, of which the last split into a western and an eastern branch at a very early stage. Although Frisian, Saxon and East Low Franconian have been partly ousted, partly strongly influenced by the central dialects of Holland and Brabant, both of which have predominant Western Low Franconian characteristics, the old distinctions did to a certain degree live on in later evolutionary stages of dialects.

Apart from the associations based on the old tribal bonds, there is another, even hazier factor, connected with West Germanic history. In his De Germania Tacitus divided the Germanic tribes into Ingvaeones, Istvaeones and Erminones, of which the former occupied the coastal regions. It is not clear if this distribution maps in any definable way on to the division based on tribal bonds given above; but the term 'Ingvaeonism' has gained some popularity in Dutch historical linguistics, with reference to the quite considerable number of characteristics (morphophonemic, lexical and even syntactic) common to a great many coastal dialects, irrespective of cata- loguing as 'Franconian', 'Frisian' or 'Saxon'.

Even more important appears to be the extralinguistic diachronic factor: the Germanic-speaking Low countries grew together into one state at a fairly late date (late sixteenth century), and almost immediately broke up again into two political entities, due to the 'Reconquista' of the Southern Netherlands by the Spanish monarchy. In fact there was, until the end of the eighteenth century, a third state, that of Liège, to which most of the (Dutch-speaking) southeastern province of Belgian Limburg belonged. After the sixteenth century the southern dialects developed independently of a unifying standard language. For centuries, even up to the 1930s, most if not all administration was conducted in French, and most education was in French and Latin (the latter at the university). Regional dialects continued to be used in everyday life, but natural developments, as well as contact with the dominant language, French, continually drove them further apart, both vis-à-vis one another and with respect to the northern dialects. To all this may be added another external one. For the Roman Catholic clergy the northern dialects, especially those of the central provinces, were associated with Calvinism. Although most of the priests were favourable towards the local vernaculars (and not towards propagation of French), they tried to stop whatever linguistic influence might have come from the northern neighbour. The evolution in the northern dialects was less turbulent, and to some degree it was mitigated by a common written language. But even there the status of Dutch as a unifying language was not always undisputed, especially in a number of peripheral provinces. In Groningen, Low German was a formidable rival for some time, and the southern province of (eastern) Limburg was not attached to the Netherlands until 1848.

The original situation of pluriformity and the external historical facts converge on a picture of extreme dialectal diversification. Traditionally the modern dialects are divided into 5 large groups […]

— König & van der Auwera/ De Schutter

Arguably, this shows that "Dutch" was not standardised enough to make one identifiable entity for a very long time, resulting in fairly large differences 'on the ground', even today, which is is not entirely dissimilar to the situation for German. This political angle
as a historical process that shapes the languages results in possible date for this answer that would put it at roughly the later middle ages, as earlier the different dialect groups would be just 'too continuous'.

This long lasting profound lack of standardisation of the language in the everyday talk of people is well attested for German as well (— Stefan Eispaß: "Standard German in the 19th century? (Counter-) evidence from the private correspondence of 'ordinary people'", in: Andrew R. Linn & Nicola Mclelland (Eds): "Standardization. Studies from the Germanic Languages", John Benjamins Publishing Company: Amsterdam, Philadelphia, 2002, p43–66.)

One early political document that might be interpreted as signifying 'one common language' for all ancestors of the three langugaes mentioned might be seen (with ample caveats) in the 'Strasbourg oaths', a treaty between Frankish rulers written and read in the two most basic and widely understood languages of the realm:

Louis the German swore his oath in Romance so that the soldiers of Charles the Bald could understand him. Likewise, the latter recited his in Germanic so that Louis' soldiers would understand.

This would put the ancestors of Dutch and German as 'close enough' for intelligibility, but the caveats are of course that this clearly differentiates just Romance/French from Germanic/German, but doesn't tell us much about the internal differences within the Germanic side, and further documents to compare this very likely internal division are lacking.

While this dating would come still somewhat close to the 'widely accepted' view in scholarship, the remaining uncertainties are massive and provide enough room for exploring rather radical alternative arguments:

The present paper gathers all the evidence and arguments known to the author to support his contention that this division occurred at least half a millennium earlier. […]

[…] three groups of arguments in favor of the view that the High Germanic Consonant Shift, and thus the division between High and Low Germanic, is not a change of the fifth to eighth centuries, which is the handbook view, but is older by several centuries. […]

Since the Ubians were moved to the area of the inscriptions in 38 B.c. and had settled in their preceding habitats before 55 B.c., and since they must have already brought the Shift to these territories, it follows that the Shift is at least equally old, namely at least as old as the first half of the last century B.c.

— Theo Vennemann: "Dating the division between High and Low Germanic. A summary of arguments", in: Werner Winter (ed): "Trends in Linguistics Studies and Monographs 73", Mouton de Gruyter: Berlin, New York, 2010. (p271–304)

Such a date for a root 'cause' for grouping the three languages into different branches, clearly distinct, is not the last word this, as the consequences are really remote to grasp, and put the emerging more direct distinction closer to the end of the medieval period:

A rare incunabulum, printed in 1482 — only 10 years before the 'discovery' of the New World — in the city of Gouda in the present-day province of South Holland, contains what is believed to be the first instance of the term Nederlands 'Netherlandic/-isch, Dutch'. It occurs as part of the formula also wel overlantsche als nederlantsche tale ende sprake "both 'Upperlandic' and 'Netherlandic' language and speech". This belief is partly right, but it is also partly wrong. It is right insofar as nederlantsch here undoubtedly refers to the form of language used in the Low Countries, on the eastern shores of the North Sea. It contrasts with the term overlantsch, for which 25 years earlier (in 1457) the synonym dat hoghe duutsche had already been used. The latter occurs in a breviary, also from Holland (Van Wijk 1910:239-240):3 "dese oefeninghe is ghetogen vanden hoghen duutsche int neder duutsche" ("this (spiritual) exercise has been translated from High 'Dutch'4 into Low Dutch").

In contrast to today, the term neder duutsch here refers not only to the language of Northern Germany, but to all the speech forms spoken along the shores of the North Sea and the Baltic Sea, from Dunkirk to Reval (Tallinn). It is interesting to note that these contrasting terms occur first in this Holland breviary rather than in some text from Germany itself. There the first reference to a corresponding distinction is a famous passage in a sermon dating from ca. 1275 and attributed to Berthold von Regensburg (Pfeiffer 1965:250/38-251/2):

Ir wizzet wol, daz die niderlender und die oberlender gar ungelîch sint an der sprâche und an den siten. Die von Oberlant, dort her von Zürich, die redent vil anders danne die von Niderlande, von Sahsen, die sint ungelîch an der sprâche.

[You may know that the 'Niderlender' and the 'Oberlender' are quite different in speech and manners. People from the 'Oberlant', from Zürich, speak differently from those from the 'Niderlant', from Saxony; they are quite different in speech.]

Of course, the term 'Oberlender' refers not only to the inhabitants of Zurich, nor does 'Niderlender' refer to those from Saxony alone; these places are merely taken to represent the larger opposition between North and South to which the statement primarily refers. Similarly, a quarter of a millennium later, Martin Luther strove to write in such a way that "mich beide, ober- und niederlender verstehen mögen," (that "both ober- and niderlender may understand me"). In my opinion, Luther might well have subsumed present-day speakers of Dutch under the latter.

In Luther's day, they were the inhabitants of the Burgundian Kreis within the Holy Roman Empire, including such important duchies and counties as Brabant, Holland, Zealand and (since 1529) the county of Flanders, for most of its part a former French fiefdom. Medieval sources never clarify quite where the frontier between oberlant and niderlant lay. Whether it coincided with the famous Benrath isogloss (maken/machen line), the northernmost limit of the territory affected by the High German or Second Germanic Consonant Shift, is rather questionable. However important from the point of view of the linguistic system, this isogloss appears to have been quite meaningless even in Old High German times (before 1000) as far as language awareness is concerned (cf. Klein 1990:40). In medieval times, Cologne was one of several places sometimes associated with the niderlant and sometimes (if more rarely) with the oberlant (cf. Schützeichel 1963).

In fact, 'languages', in the modern sense of autonomous, standardized constructs used for communication and symbolism by well-delimited speech communities, were seldom recognized by medieval speakers. Instead, the contemporary view held that Western Europe was fundamentally divided into Romance and Germanic, as expressed succinctly by the Brabantine poet Jan van Boendale (d. ca. 1352):

Want tkerstenheit es gedeelt in tween Die Walsche tongen die es een Dandre die Dietsche al geheel.

[For Christendom is dealt in twain / the 'Welsh' tongues being one / the other, 'Dietish', all and hale.]

— Luc De Grauwe: "Emerging Mother-Tongue Awareness The Special Case Of Dutch And German in the Middle Ages and the Early Modern Period", in: Andrew R. Linn & Nicola Mclelland (eds): "Standardization. Studies From The Germanic Languages", John Benjamins Publishing Company: Amsterdam, Philadelphia, 2002, p 99–116.

One final but especially striking example of the problems arising from the deplorable scarcity of evidence is found when trying to analyse Old Dutch in its earliest surviving examples:

The following incantation, meant to cure horses, came to us from the end of the ninth century (Van der Sijs & Willemyns 2009, 153):

Visc flot aftar themo uuatare.
uerbrustun. sina uetherun.
tho gihelida. ina. use druhtin.
the seluo druhtin. thie thena uisc gihelda.
thie gihele. that hers theru. spurihelti. AMEN

Translation: “a fish was floating over the water with broken fins. Then it happened that our Lord cured him. May the Lord, who cured this fish, also cure the paralysis of this horse. Amen.” Today we believe it to be Old Dutch, but not so long ago many scholars assumed it to be Old Saxon (Old Low German).

This proves not only that we know less than we would like about the older stages of our languages but also that the difference between them is not so easy to determine and that the boundaries were rather frayed. Even nowadays the dialect of a citizen of Venlo (in the Dutch province of Limburg) may be hard to distinguish from that of an inhabitant of Straelen (the first village on the German side of the border). And, in fact, why do they have to be distinguished? What is actually spoken there are two almost identical language varieties that happen to be used in two different countries and that only scholars want to categorize as one being Dutch and the other being German […]. Today and—with even more reason—15 centuries ago, the speakers themselves quite rightly don’t and didn’t bother.

— Roland Willemyns: "Dutch. Biography of a Language", Oxford University Press: Oxford, New York, 2013. (Illustrating in a very accessible way that paucity of evidence and lack of firm knowledge can still give insights and entertain. The later stages of development stnd on much firmer ground though. To understand why this question has such a lengthy and seemingly overly complex answer, this is the first recommended book to read in its entirety.)