There is a particular timeline I've considered writing for a while, it would be my first one.
I first learned of the incident in question several years ago in high school while looking through a book that detailed the travels and origins of prominent Middle Eastern and Chinese explorers. One of the most fascinating stories was about a fellow named Rabban Sauma.
He was from a Nestorian family, born in 1225. A monk, he chose to travel with one of his disciples to the holy Land. Only making it as far as Baghdad due to Islamic control of his destination he settled down for a few years.
At this point some important events happened. The same disciple that he travelled with became the patriarch of the Nestorian church, giving Sauma close ties to a prominent religious icon (several khans at the time and many other Mongols were Nestorian christians), and Sauma gained quite the reputation as an explorer.
In 1287, he was commissioned by the khan of the Il-khanate to become an envoy of sorts to Christian Europe, hoping to gain an alliance against Egypt. Sauma met with bishops in Rome, Philip the Fair of France, and King Edward I of England.
Now, the main reason that Britain and France did not subscribe to such a coalition against Egypt was due to fighting between the two countries. Indeed, both Kings told Sauma that the reason they were unable to commit to such an act was due to conflict between England and France. The various european rulers including Pope Nicholas IV wrote letters back to the khan of the Il-khanate and gave vague answers, saying they couldn't gurantee an alliance.
Now, say that the Europeans were able to settle these differences. What if the pontiff and Christian rulers of Europe joined forces with the Il-khanate to drive Islam out of the Middle East? Assuming that a victory was possible, Islam might have withered significantly with the fall of Egypt and Africa would be opened to the combined European and Mongol forces. Mongol culture may have spread even further. And in the end, could we even see a sort of cold war with the Europeans and Mongols playing the parts of medieval superpowers?