I think a lot depends on the terms of the final settlement. The treaty might insist on a special status for the city, and as such, some protections for places of worship for religious minorities.
I think it goes without saying that every surviving church in the city would be restored to the Patriarchate of Constantinople, starting with Hagia Sophia (which would require a massive, long-term restoration). There would undoubtedly be pressure to take over some or all mosques, no question.
It is instructive, however, to look at what happened to other cities in the Balkans (especially Greece) where large Muslim populations lived at the handover to the new Christian states. Mostly, they fled, or were exchanged. In Thessalonica, the
New Mosque was not marked for takeover, but the departure of the city's entire Muslim population left it defunct, and it ended up as a museum. How many Muslims would want to remain in a Greek-ruled Constantinople? Even without a formal requirement of population exchange for the city? My guess is that more than one mosque would end up defunct, at least, and converted to other purposes.
It's a fair question what would become of prominent edifices such as the Blue Mosque in such a situation.