If the Ottomans are partitioned I could see this happening or them performing worse in the Balkan Wars(one of these happening somewhere between 1900 and 1914). Greece, Bulgaria, Russia, or a mix of the 3 are probably your best bet on doing this. Maybe Russia works more with Bulgaria. Bulgaria keeps the majority of Thrace and also keeps its Greek lands. Greek is given its claims over Anatolia as a way to make them go along with Russian and Bulgarian gains. Due to Britain refusing to accept direct Russian control over the strait and the city of Constantinople both sides come to a compromise. The city is made into a independent city state under the Orthodox patriarch and church. Russia agrees to it because they think a theocratic Orthodox state would support Russia who is the biggest Orthodox power and near reactionary. Pontus and Armenia are made into independent countries out of former ottoman territory while the rest of the empire is divided up by European powers(imagine allies originally plan for the Ottomans after WW1). This prevents any Turkish state from ever taking back the city. The Orthodox city state probably follows the example of some of the Orthodox nations in the Balkans and force out almost all of its Muslim population. This might be counteracted by British and maybe even Bulgaria treaty agreements that guarantee the protection of these people. But even then the Muslim population could still drop greatly. I doubt they would tolerate living under a theocratic Christian city state who likely treats them poorly. The city state could be a mix of Greeks, Russians, Bulgarians, and other Slavs. Best case for Muslims is a situation similar to otl Bulgaria were they are somewhat tolerated due to them still being a large minority even after being pressured to leave. The city state under the Orthodox Church is likely divided politically between supporting Russia, Bulgaria, or Greece. Maybe it becomes the Geneva of Eastern Europe. Serbia, Russia, Greece, Romania, Pontus, and Armenia use the city as neutral ground for diplomacy. Eastern Europe was a lot more reactionary or to use a nicer word traditional before communism so I don’t think some of this is too much of a stretch. The only issue is Western Europe not accepting this.
A second option but unlikely one I think could be interesting is a more successful and powerful Italian nation taking the city. Maybe Italy becomes more involved and is able to prove themselves as a major power in the Balkan Wars or WW1. Britain allows Italy to take the city due to them not fearing the Italians and just to screw over Russia. Britain in this pod doesn’t think a independent city will last and don’t think the Greeks are strong enough to take it or hold it. Britain also sees this as a good way to make the Italians more friendly with them and hoping it pulls Italy away from its German leaning. The ironic impact of this might be Britain unintentionally worsening their relations with France who might support the Russians and get upset over Britain helping strengthen a possible French enemy and German ally which could lead to Britain becoming more favorable towards the central powers. I imagine Italy taking Constantinople would probably be considered the greatest national victory since unification which could have a big impact of public mindsets in the country. But I could see a Italian controlled Constantinople trying to unite both churches as a propaganda symbol and statement. If the Italians come to a agreement with the Catholic Church similar to the one in otl I could see a similar agreement being made with the Orthodox Church who is now officially united with the western church but in practice functions much the same as before. Maybe something similar to Eastern Catholics in Austria-Hungary is encouraged by the Italians. The Eastern Church still believes in the same stuff, has the same structure, and ceremonies are unchanged but the only difference is now they are in communion with the western church and self identification is different. This is only recognized among Catholics, Greeks, and Orthodox Albanians. Maybe even Romanians even come around to recognizing it. Orthodox Slavs consider the unification as illegitimate and don’t recognized it but that is unimportant if those nations still eventually fall to communism. Like the other pod Italy probably becomes closer to Greece in this pod and helps them take their claims over Anatolia and maybe helps push for a independent Pontus so the Greeks look the other way about Italian rule over certain Greek claims. But this process slowly makes Greece more dependent on Italy to the point of becoming a near client or puppet state. Same goes for Pontus who has to deal with a large discriminated Turkish population in it. Britain supports Italy in the region seeing them as a counterbalance to Russian influence in the region.