John Reilly (in characteristically controversial but amusing fashion) suggests an alternative possibility: that the West and the Mongols might have cooperated in a joint attack on the Islamic world. He is reviewing Steven Runciman's classic History of the Crusades. Reilly's own assumptions are shaped by his conservative Catholic position.
One of the impressions I took away from this history was that the legend of Prester John was essentially correct. The Mongols were not themselves predominantly Christian, but their neighbors and onetime overlords, the Kerait people of northeastern Asia, were Christians of the Nestorian variety. The Mongols absorbed the Kerait territory and leadership, as well as their script. Although the Mongol empire, at least in the beginning, made a point of religious tolerance, there was heavy Christian influence among the advisers of Genghis Khan. There was even more Christian influence on Hulagu, who conquered southwestern Asia and eventually became the Ilkhan of Persia.
In the 13th century, there was quite a lot of coming and going between the principal courts of Europe and those of the khans, all with an eye to coordinating an attack on Islam. European representatives to the Mongol capital at Karakorum were exasperated by the Mongol principle that there were no sovereign states in the world, only current and future vassals of the Great Khan. Nonetheless, encouraging words were exchanged. Rather more substantive talks took place with a Nestorian priest from Hulagu's court, who said Mass for Edward I of England and received communion from the hands of the pope himself.
None of this really came to much. The Franks of Outremer were actually more cautious about allying with the Mongols than were their cousins in Europe. It was the local Christians, the Armenians and the Georgians, who accompanied Hulagu when he destroyed Baghdad in 1258. Few world cities have been as thoroughly destroyed as Baghdad was. While they were at it, the Mongols slew the Caliph, who had surrendered to them, and ended the Abbasid Caliphate, along with the Old Regime of Islamic history.
Runciman does allow that Outremer might have made a decisive difference, if it had cooperated with the Mongols. Though he restrains himself from elaborating the what-ifs, he hints that, in the 13th century, it might have been possible to end Islam as we know it. With some Western aid and encouragement, the Mongols might have become the ruling stratum all the way to Egypt. It is likely that they would have embraced some form of Christianity, then the religion of large minorities in the Muslim world, and in some places even majorities. Islam, with Baghdad in ruins and with Mecca and Medina soon to follow, might have shrunk to the status of the heterodox Christian churches of the Byzantine Empire. There are problems with this scenario, but not so many that it could not serve as the premise of a counterfactual novel.