Basically nothing happened. The take over was a power and money issue and not ideological or religious. Since New Netherlands was at that time the biggest and most profitable of the american colonies they left basically everything as it was. Practically only the governor was replaced. Besides the people living in the New Netherlands were not only dutch. At that time it was already a mix of nationalities living there. just like New York nowadays. There were already people living there referring to themselves as Americans and not Dutchmen, noticeably mr Van Rensselaar.

A good description of these times is the book by Russel Shorto "The Island at the Centre of the World", very readable, informative and entertaining.