A lot depends on how the Islamic powers react to the events as described.
Of course, then there's the Mongols. I've raised the notion that, since they are operating at the limits of their strategic range once they get to the Levant and Anatolia, the Christians (being well-organised a this juncture) can withstand them of they act in concert. This would result in a single Ain Jalut-style defeat for the Mongols, after which pretty definitive borders are drawn up by default.
This means that I expect the Muslim powers to get absolutely hammered from the East, while the Christian powers take a major beating, but keep their footing. And then they have the Ilkhanate next door. What happens to Islam depends on the Ilkhans. If they convert to Islam, it prevails. But Islam is already weakened a lot by this point. So I can easily see the Ilkhans converting to Nestorianism (as two of them did in OTL). Thus, you'd see Islam getting wiped out. Maye some small remnants would continue to exist, but it would cease to be a major religion.
The more recent you get, the more difficult it becomes. I'm sure there are people who can come up with some specific examples where things could be done. The last really major opportunity I'm aware of is the Russian desire to go "back to Byzantium", particularly in the latter stages of the 18th century. You have to really steer thing so they work out for Russia and is allies (in the event, that would be Austria), but it can be done. And then you can have them carve up the Ottoman Empire, with particular attention to rewarding all the Christian minorities within the empire with their own brand-new states. This would involve:
-- A big-ass Greece, a.k.a. "the Byzantine Kingdom", eventually to be placed under
Konstantin, with a regency first. (Some forced relocation of Turks required.)
-- A big-ass Georgia. (Much less relocation required.)
-- A big-ass Armenia. (More relocation of Turks, and also Kurds.)
-- A small Assyria. (Because there's no way to make it any bigger than small, and even they you need to kick out some Kurds and Arabs to make it work.)
-- A Levantine state for (primarily) Maronite Christians. (... a.k.a. Lebanon, but everyone who's not a Christian gets three days to leave before they're put to the sword.)
-- The ultra-shiny mega-holy Kingdom of Jerusalem. (Muslims and Jews will be forced out. Expect joint Catholic-Orthodox oversight over this state.)
-- Maybe something for the Copts in Egypt, if that can be made into a workable project. (Press X to doubt.)
Beyond that, Arabs and Kurds will all be going their own way, while the Turkish heartland (central Anatolia) ends up surrounded by Greeks, Georgians and Armenians (who all hate them) backed by (primarily) Russia, who will not hesitate to reward any sign of Turkish revanchism with full-scale genocide.
So, yeah, that gets it done, too. But as you can see, it's a messy affair with loads of little wars and atrocities scheduled for the near- to mid-term future. And the later we go with the POD, the worse this gets. Do European powers in the modern period have the stomarch for this? Rule of thumb: the closer to home, the less they're willing to be truly brutal. So I'd say "no".
The only reason Russia would be able to really get away with what I'm suggesting here in the later 18th century is because we can time the whole thing to perfectly line up with the great French republican cock-up. As in: Russia and Austria dismantle the Ottomans together, and then events in France force Austria to turn all its attention West, and Russia gets a free hand to carve out Christian client states by any means necessary.
(The motive for Russia to go that far is that Russia's real aim is to unite Russia with the proposed Byzantine Kingdom, thus making good on Russia's claim to have rightfully inherited the Byzantine legacy. Third Rome, third Rome!)