First, I have to point out that the Rambam, although he lived in Egypt, was no fan of Islam or Mohammed who he called a "madman" and "the defective one" in his Letter to Yemen.
On the general point: Over the last 2000 years, Jews have migrated extensively; probably more so than any other nation or people. Here is a nice summary of the post-exile movement of our people. As you can see from the descriptions and the map at the bottom of the page, Jewish migration was often linked to economic potential and cultural advantages and disadvantages. Therefore a variety of issues caused some people to emigrate to Rome, others to Greece, east to Babylonia, Europe or along the African and Asia Minor coasts.
For the first six centuries, the Christians were not a major problem to Jews as the Christians were busy battling and converting (often by force) the pagan tribes of Europe. Dimont, "Jews, G-d and History, pp. 151-152. Also Jews benefited because they were more educated and cultured than the newly-converted pagans were, and rose to relatively high positions. Ibid. pp. 213.
During the feudal era, Europe held some attraction to Jews because of the restrictions placed on Jews. Christians were tied to the land or if they were craftsmen, they were tied to their guilds and forbidden by law to lend money at interest. Jews, however, were forbidden to own land or join the guilds. But that meant they had nothing tying them down, and they alone could lend money with interest to non-Jews. Carroll, "Constatine's Sword: The Church and the Jews: A History," pp 243-44.
The conflict between Moslems and Christians, during the Crusades period, also benefited Jews who lived in or traveled to both worlds. Welcome in both worlds, Jews were able to sell the goods of Moslems (and others through Asia) to the Christian nations, thereby pollinating Europe with many influences from the Orient. See Abrahams, "Jewish Life in the Middle Ages," p. 214; Stillman, "Jewish Merchants," contained in Meri and Bacharach, "Medieval Islamic Civilization L-Z" at pp 497-498.
But with the success of Jewish merchants came a downside: "When Feudal man realized the superiority of the Jewish way of doing things, he absorbed Jewish know-how, kicked the Jews out to eliminate the competition, and went into business for himself." Dimont, at 224. Jews migrated instead to Eastern Europe where their special skills were welcomed. Ibid.
Jews did well in the Islamic world, depending upon the rise and fall of various sects. Jews during Rambam's early years were respected and protected in Andalusia during the reign of the Almoravids, but when they were deposed in 1148 by the Berbers, that sect forced Jews to convert to Islam if they chose to stay. 1954 Encyclopedia Americana, vol. 18, p. 140.
Similarly, other Jewish communities in the Arab world suffered ups and downs depending on who was in charge.
But, as always, Jews outside of their homeland live where they are welcome, and leave where they are not. Before World War II, 9.5 million Jews lived in Europe. Holocaust Encyclopedia. But because of the Holocaust and continued anti-semitism in parts of Europe, only 1.45 million reside there, according to the American Jewish Yearbook for 2010. The Arab-Israel conflict has caused a sharp downturn in the Jewish population in Moslem countries. The Jewish population in the Middle East dropped from 1 million in 1948, including Iran but excluding Israel, to less than 20,000 in 2010. That decrease includes 867,000 Jews who were expelled, or forced out of Moslem countries at war with Israel. "The Forgotten Exodus," Jerusalem Post Special Supplement, Nov. 29, 2002, pp. 12-13.
To summarize: The fact that Christians are or are not idolators is not determinative of where Jews live. They live where it is in their own interests.