Did the pope choose sides between the Allies and the Axis in World War 2?

No, he did not, and it has cost him and the Catholic church a lot of moral leadership and capital since then.

The Wikipedia article Pope Pius XII and the Holocaust is very thorough on this topic actually. However, I want to explore two issues that it only briefly touched on.

One is the protestation of the Pope that any condemnation he could have made against the treatment of the Jews would have only made their plight worse. However:

• 6 million Jews were killed during the holocaust.
• Besides the death camps which most people are familiar with, there were slave labor camps which had even worse conditions and fatality rates—in the hundreds per day.
• There were more than 15,000 (some think as high as 20,000; we may never know the exact number) death/slave/prisoner camps during the war.
• Beyond the 6 million Jews killed during the Holocaust, 5 million more people died, including 1 million Catholics from Poland.

From these facts its very hard to see how the condition or plight of the Jews, or any prisoner of the Nazis', could have gotten any worse.

It's also important to note that WWII brought about a significant change it the way that Jews were regarded throughout the world, particularly in Europe. Antisemitism isn't a new thing, and it wasn't a new thing during WWII. At various times the Catholic church even endorsed it.

I'm in no way saying that the Catholic church, or Pope Pius XII, endorsed what Hitler did, but it needs to be pointed out that the world view of Judaism was very different before and after WWII. This difference indeed contributed to the silence and timidity of the Vatican on denouncing the actions of the Nazis.

It's also important to note that the Pope, while timid, was a Jewish sympathizer and was afraid that the actions he took would engender a German retaliation. Because of this he made plans for his own de facto resignation should he be taken prisoner and for the movement of the Holy See to Portugal.

Finally the Vatican has throughout the 19th and 20th centuries tried to maintain a stance of neutrality on international conflicts. Ultimately whether you agree with this or not (it's criticized even today), the actions of Pope Pius XII can be seen as a continuation of this policy and a strong mental fortitude to do what he perceived to be the right course of action.

When asked this question, I do like to point to one little-examined incident in 1942. In July of 1942 the Dutch bishops actually did send a letter to the Nazi government protesting, specifically, the Nazi treatment of the Jews.

Ours is a time of great tribulations of which two are foremost: the sad destiny of the Jews and the plight of those deported for forced labor. … All of us must be aware of the terrible sufferings which both of them have to undergo, due to no guilt of their own. We have learned with deep pain of the new dispositions which impose upon innocent Jewish men, women and children the deportation into foreign lands. … The incredible suffering which these measures cause to more than 10,000 people is in absolute opposition to the divine precepts of justice and charity. … Let us pray to God and for the intercession of Mary … that he may lend his strength to the people of Israel, so severely tried in anguish and persecution

Now, as a respectful, thoughtful group of individuals, what do you think the Nazis did? Did they stop sending Jews to the gas chamber? ... I don't think so. In fact, this letter corresponds to a dramatic increase in the rate of detainment and deportation of Dutch Jews including one, now canonized, Edith Stein.

Now, admittedly, this was in 1942, so it clearly does not excuse silence from before that time, right? I'll agree wholeheartedly, except that I'll point out that Pius XII wasn't silent. Quoth the Third Reich's Chief Security Service:

Pacelli has already made himself prominent by his attacks on National Socialism during his tenure as Cardinal Secretary of State, a fact which earned him the hearty approval of the Democratic States during the papal elections. . . . How much Pacelli is celebrated as an ally of the Democracies is especially emphasized in the French Press.

(And I think it also worth noting that Pacelli was consecrated Pius XII in March of 1939, ten years after the Vatican was subjected to Mussolini's government, and five months before Hitler invaded Poland.)

But let us look to those who have been persecuted, and see whether they felt that Pius XII supported Hitler. Quoth the chief rabbi of Jerusalem:

The people of Israel will never forget what His Holiness and his illustrious delegates, inspired by the eternal principles of religion which form the very foundations of true civilization, are doing for us unfortunate brothers and sisters in the most tragic hour of our history, which is living proof of divine Providence in this world.

And then there are the numbers:

instrumental in saving at least 700,000 but probably as many as 860,000 Jews from certain death at Nazi hands

(Just in case anyone is counting, Schindler is at ~1,200. Pius at > 700,000. Pius > Schindler)

So, what do I see? Pius XII preached against Hitler, but he realizes that his flock will be put to the wolves if he preaches too readily. So, instead, he works behind the scenes to keep safe as many as he can.