Wealthy survivors didn't have much use for hospitals - their families generally arranged for private care at home or in hotels. 3rd Class and crew survivors in need of medical attention were mostly taken to St Vincents Hospital in New York, and some remained there for weeks or even longer, but you'd need to look at each individual case to be sure of the situation.
I am very interested also to find info regarding the hospitals. My g/grandfather George Combes/Coombes was a survivor and taken to Hospital. But I believe that he was transported back to England and died there not too long after the trip. So hard to find what you need. Any help will be appreciated.