What did the Roman Catholic Church do in Europe while WWII went on? Back in the Middle Ages, the Pope would snap his fingers and the crusades would start.
This is a good question. There is a world of difference between the Middles Ages and the middle of the twentieth century. The Catholic church lost a lot of power and influence over time. Think of the Protestant Reformation and the Enlightenment. Moreover, Germany became mostly a Lutheran church, and so Catholicism weakened considerably. In light of this the pope could not sway nations.
Also it should be said that Catholics fought on both sides of the conflict. Perhaps this was one of the reasons that the church technically remained neutral. I say technically, because Pope Pius XI did speak out against the evils of the Nazis and shared intelligence with the allied forces. Another point worth considering is that Rome was under the power of the Nazi, but Vatican city was never occupied.
Pius XII did three things. First, he called world leaders to prevent a war and seek peace. Second, in his first encyclical, he remained neutral, but he helped the allied forces and most importantly the German resistance. Third, in 1942 he spoke out against Nazi crimes against Jews.
Here is a sample of what Pius wrote:
The blood of countless human beings, even noncombatants, raises a piteous dirge over a nation such as Our dear Poland, which, for its fidelity to the Church, for its services in the defense of Christian civilization, written in indelible characters in the annals of history, has a right to the generous and brotherly sympathy of the whole world, while it awaits, relying on the powerful intercession of Mary, Help of Christians, the hour of a resurrection in harmony with the principles of justice and true peace.
He also used the Vatican Radio to speak out against atrocities. Here is another example of what the Pope said:
In accordance with these principles of equality, the Church devotes her care to forming cultured native clergy and gradually increasing the number of native Bishops. And in order to give external expression to these, Our intentions, We have chosen the forthcoming Feast of Christ the King to raise to the Episcopal dignity at the Tomb of the Apostles twelve representatives of widely different peoples and races. In the midst of the disruptive contrasts which divide the human family, may this solemn act proclaim to all Our sons, scattered over the world, that the spirit, the teaching and the work of the Church can never be other than that which the Apostle of the Gentiles preached: "putting on the new, (man) him who is renewed unto knowledge, according to the image of him that created him. Where there is neither Gentile nor Jew, circumcision nor uncircumcision, barbarian nor Scythian, bond nor free. But Christ is all and in all" (Colossians iii. 10, 11)
In conclusion, the Catholic church did what it could do, but it did not have the power to make a huge difference.
You can find the first quote from Pope Pius XII in the book The Holocaust, the Church, and the Law of Unintended Consequences by Anthony J. Sciolino. You can find the second quote on this site: https://w2.vatican.va/content/pius-xii/en/encyclicals/documents/hf_p-xii_enc_20101939_summi-pontificatus.html. This is the official Vatican homepage and it includes an archive of papal documents.
Unlike the Middle Ages, Pope and the Roman Catholic Church did not have much influence during the World War II. In fact, United States (one of the superpowers of the era) did not even have any diplomatic relations with the Vatican for a period of seventy years, prior to Poland's annexing by Germany. Similarly, Hitler wanted a single Reich Church, a church controlled by him. Britain was not too fond of church either, France (under Mussolini) was thought of as a supporter of the vatican. Overall, the church had a very weak position.
Pope Pius XII did all he could to maintain the neutrality of Rome in the wake of the war, however, refrained from actively denouncing Nazi atrocities fearing backlash from Hitler. He used Vatican Radio and other channels to promote peace and helped jews escape the Nazi. But other than that roman catholic church was a mute spectator to the world war II, having lost most of its relevance.