While the Mongols will use Christians in administration and governance, I find it hard to believe they'd either convert to any Christian sect (be it Copticism or Nestorianism), let alone give Buddhism more power (even if there will be at least a few Buddhists in Alexandria again for the first time since Antiquity). They aren't going to succeed in conquering Egypt without the support of at least some emirs who were pro-Mamluk OTL. Further, ruling Egypt means they'll have to come to terms with the Crusader sponsorship of piracy, and that probably means at least one Mongol-Crusader war.
I'm curious what would happen to the Abbasid caliph in Cairo. I'd assume he'll be deported back to Baghdad to remain as a pillar of legitimacy for the Ilkhanate. Since several times there were proposals for an expedition to capture Mecca and Medina, we can assume this would also be a Mongol campaign. I definitely cannot see the Ilkhanate doing anything but converting to Islam should they grab those cities, since it's far too beneficial among the people with actual power in their empire.
Problem is this structure is very unstable. They already some problems governing Anatolia OTL since their governors there would take independent action contrary to what the state desired (i.e. the one Mongol general who murdered the Armenian king to strengthen his position among local rulers, or Timurtash who outright rebelled at one point) and Egypt gives one hell of a powerbase. Dealing with this draws attention away from critical frontiers in the north and east against the Jochids and Chaghatai Khanate and means some tribal allegiances might shift in the wrong direction--OTL this hindered Ilkhanate invasions of the Levant.
10,000 Mongols are a very significant military force in this period. And Egypt has historically been conquered by small military forces; the Mamluks are incredibly unpopular, so there probably won't be any great rising of the Egyptians against the Mongols. Once the Mamluks have been expelled, the Mongols can pretty much just take over the existing administrative structures.
The Mamluks were surprisingly resilient. The first decade of the 14th century they suffered a huge defeat against the Ilkhanate and lost Damascus, faced a series of crusader raids, faced rebellions from over a dozen emirs, dealt with uprisings among coastal Bedouins as well as the Druze and Maronites, and had a short but violent succession struggle, yet the state ended up defeating all of this and Egypt remained prosperous (the Levant did not thanks to the piracy and Mamluk neglect).
Hey, I'd say they're going to conquer Anatolia as well. The Ottomans will be vassalized. In the long run this will probably delay the Ottoman rise for a long time, with the Turks having to deal with the expulsion of the Mongols from Anatolia first. The Ilkhanate may inherit Egypt, but it is also possible to have another Mongol khanate. We probably have a Mongolian elite in Egypt. Regarding the economy, I don't know if they can be like the Mamelukes and continue exploring the region or try to make the region prosper. In relation to the Ottomans, if they ascend. I tend to think that they will be more resistant than the mamelucos. Which makes the conquest of Egypt unlikely or more difficult. This will also make the Portuguese turned the entire Indian Ocean into a Portuguese lake. Only the Ottomans managed to compete with them in this region. So then the Red Sea, the Persian Gulf, and the Indian Sea will be completely dominated by the Portuguese.
Oh and the commander of this invasion did not like Muslims and preferred Christians. So they will probably be used as enforcers of the Mongols.
I'd say the Ottomans are a non-entity TTL since they'd be the victims of someone like Timurtash who manages to subdue them and as government authority weakens, may well establish an independent state (as Timurtash's chief lieutenant Eretna later did) that could finish subjugating the Anatolian beyliks and either become another Sultanate of Rum or follow the more ambitious path the Ottoman Empire did OTL.
Ilkhanate Egypt will probably not be as prosperous as OTL since a lot of its trade will instead go through the Levant.
What happens in Egypt after the Ilkhanate loses control (and they will, they will be lucky to survive the Black Death and the droughts of the mid-14th century), I'm not sure. Maybe another Mamluk Sultanate gains power, maybe another crusade will sweep into the region and make Egypt a Christian kingdom, or maybe the post-Mongol state will prosper. It could be an Ilkhanate remnant akin to the Chaghatai ones in Central Asia, or it could be something like the Timurids or Jalayirids where a powerful emir rules in the name of the khan but the khan has no actual power and eventually the khan is cast aside and the emir mints his own coins.
As for the Portuguese, neither the Mamluks nor Ilkhans had much of a naval tradition to speak of, so while they'll very likely have a Red Sea fleet as the Mamluks did, they'll outsource their naval policing needs to the Crusaders or Italian city-states.
