It's almost impossible to consider an Orthodox country conquering Rome after 1900. It would basically require all of the powers in Western Europe and the US to stand idly by while...Russia? Some sort of insane Greece? conquers one of the most historically significant cities in the West. Even Protestant countries would be flocking to Italy's aid, to say nothing of the Eternal City itself.
While the other cities wouldn't see such widespread reactions, don't forget that in 1900 Alexandria is under a British protectorate, and Istanbul is the capital of a world power. An ailing one, maybe, but still plenty strong when not being opposed by other world powers (or which the only Orthodox one is Russia, though I could see England and France and Germany backing a play by Greece, maybe). Jerusalem and Antioch (Anyakya, to give its modern name) shouldn't be a problem, though - no one really cares about them in 1900 (indeed, no one really cares about Antakya to this day, and aside from the brief spat in which Turkey annexed Hatay, I don't think the city's excited any kind of interest in centuries).\
But let's wave our hands and say it does happen.
The Pope would probably a lose a little legitimacy, but the rifts between Catholic and Orthodox Churches are so deep that I can't imagine any real number of Catholics turning to Orthodoxy. The Pope goes somewhere else - maybe Avignon, if France will have him, maybe somewhere else in Italy if not the entire city falls, maybe somewhere in South America...or, whatever hypothetical Orthodox power conquers Rome could let him stay in place (as, indeed, they will if they're smart).
Protestants...at most, some will probably just take it as a further sign of the corruption and unworthiness of the Catholic Church. By 1900, almost all mainstream Protestant churches have been independent of Rome for more than 3 centuries; it's hard to imagine them being overly affected.