I do suggest that instead of the Crusaders taking Egypt as is often suggested in this thread, why not the Mongols ally with the Franks to take Egypt. The Crusader states managed to survive till the Mongol expansion even if much diminished and the Mongol general in charge during the battle of Ain Jalut was a Nestorian Christian and so was much of his Tribe(Naimians). Although he was left with just a force of about 10,000 and that was in responsible for their defeat to the Egyptians, they already had the vassalage of several Christian Kingdoms (Cillicia, Georgia and Antioch) while 2 of these won't be independent initially ITTL, the Mongols smashing into Anatolia would either destroy them or leave them independent or at least autonomous from the Romans and subject to Mongols Influence. A Monk probably Nestorian, Orthodox or maybe even Catholic being among a delegation sent to what is left of Jerusalem could inform the of their direct overlord (The Naiman General) being Christian and get their support with speculations that he is either Prester John or his Vanguard it would be better if this happens after the Egyptians camped outside Jerusalem so that they smash into the back or flanks of the Egyptian Army, saving the Mongol-Naiman force and allowing the Jerusalem high standing with the Mongols when they move to take Egypt and further expand across North Africa with Crusader, Ilkhanate and Naiman troops as the Ilkhan returns from Mongolia. Story of the Holy land, their relationship to the ruling families and relative stability from the civil wars between the Khanates could lead to an influx of Naimans from Mongolia but I don't know if the Ilkhanate would allow this, fearing consolidation against them and i also don't know if the Ilkhans would allow the the Naimans to mostly do their own thing or constantly request troops from them to fight their wars against the other Khanates but all in all this could allow for Crusaders to quickly join in taking North Africa as Naiman forces, preoccupied helping with the Ilkhanate would need extra and trustworthy troops that are preferably not entrenched in Mongol politics while the Crusaders would be happy with pledging allegiance to "Prester John". I do think Europe would for decades continue believing the Naimans have some relationship to Prester John as a letter for alliance to the Ilkhan from France talked badly of the "mohamadians" and an alliance possibly against the Saraceans even when the Ilkhans had converted to Islam. I also think the Naimans would be able to take part of Nubia as even though the Broken landscape and Hot Desert Climate would make this difficult, the Nubian states should be real weak by now but the climate still horrible for the Mongols won't go far. I expect this Naiman Horde to collapse much like the other Khanates and replaced by Franco-Naimans or Romano-Naimans. It is also entirely possible that due to the centralizing nature of the Nile River and influence from the Latins and Romans(whose capital and palaces would no doubt stun the Naimans) could adopt enough Roman and Latin influences as well as Berber cooperation so that their "collapse" would simply be them losing all non Egyptian holdings, with the Europeans and resurgent Islamic powers claiming North Africa, Jerusalem expanding into Syria and possibly Mesopotamia and a Resurgent Nubian Dynasty claiming lower Nubia.
Lastly given how Nomadic peoples tend to integrate other Nomadic peoples in their Empires better than the settled peoples we could see the Naimans do much better in integrating and Converting the Berbers and maybe instead of a collapsing Naiman Horde we just see the Naimans replaced by a more competent Christian Berber-Naiman Dynasty that can play to the Naiman, Berber, Local Christian and Crusader interest in Egypt (I expect many Europeans would migrate to Egypt as it is conquered by ally of Jerusalem and the Naimans with only a relatively small force of about 10,000 would happily take in large numbers of Europeans)
Edit;-I also do think this is one of the better scenarios a Coptic ascendancy as even though the Mongols were tolerant of all religions there are records of the Christian wives of Mongol leaders causing them to spare Christian lives so i would expect some general preference for Christians of all types in administration even if it is not institutionalized into law
While i do not know much on Mongol history from the little i have read it seems the title of Khan became related to being a descendant of Genghis Khan in a similar way that the Islamic title of Caliph became attached to the relationship of a leader's bloodline to Mohammad. This may either mean that sooner or later a descendant of Genghis Khan from the Ilkhanate is installed as the leader of the Naiman Horde or fearing the legitimacy and potential successionist fuel that a legitimate Khan can give, the Ilkhans try to make sure that the Naimans are never ruled by a decendant of Genghis Khan.
I do expect that as the Ilkhanate becomes Muslim and the Naimans move closer to Orthodoxy there would eventually be a falling out which i hope in some butterflies sort of way end up with Timurlane invading China like he wanted to do before he died
