There were and still are Orthodox patriarchs outside of Constantinople: for instance, the Patriarchate of Moscow was established in 1589.

If you mean after 1453, Gennadios was made patriarch by Mehmed II, and by all accounts they had a fairly tolerant relationship. The sultan didn't much step on Gennadios's toes after the conquest, and while the Greek population was much lessened, it seems that the pair had a fairly easy coexistence.

If you'd like to look up more about him, his birth name was Giorgios Scholarios. He had quite the life.

There was a Latin Patriarch of Constantinople, established after the western crusaders had taken the city during the Fourth Crusade, and the office, although it was basically just an honorific, persisted through the fall of the Byzantine Empire and into the 1960s.