New York city was the capital when Washington was sworn in as president. It was decided when the constitution was drafted that no state would host the permanent capital. So, NYC would have been turned into a federal district, The District of Franklin maybe. Its government would look like the District of Columbia. A mayor who would be treated as a Mayor/Governor and a City Council/ State Assembly. I can see the boroughs being treated as counties. It would be under Federal jurisdiction meaning that Congress could overrule the local government if needed.