Hi, we moved here last summer. Do you have a relocation company to help out as ours were fantastic?
I don't know what your relocation package is like, but Chappaqua is lovely and has quite a few Brits (plus the Clintons ) as the schools are fantastic. It is about 45 mins to Grand central.
Scarsdale has great schools but there are a lot of alpha moms and that put me off a lot. Katonah is gorgeous and you get stacks of property for the same money, but is a lot further from the city.
We looked at Mamoreneck and Rye, which were lovely but didn't have any rental properties when we were looking.
In NJ, Summit is lovely. That is where we nearly went, but we got more house in Westchester (and have 4 kids so needed more house!). I personally didn't like Short Hills, but other people rave about it.
We also considered Manhasset in Long Island but our relocators thought that Brits tend not to settle as well on LI, plus the traffic is pretty bad (although bad traffic here is nothing compared to the UK!!)
I am reasonably new to Mumsnet. I joined ages ago, but haven't really posted, but if I can help at all, feel free to email me.