There is a high probability, due to several factors, but most prominently the general decline of the Dutch merchant empire during the 18th century, the demographic superiority of the British and their strategic position North as well as South of New Amsterdam, that at some point before 1776, New Amsterdam would have been gobbled up.
It might then end up with a different name but New York (which appeared out of the blue during the first conquest by the British). Also, the longer it stays Dutch, the more acknowledged and cherished their heritage might be in the centuries hence.
Have you read Sholto's "Island in the middle of the world"? Definitely a good read.