If a Brit moves to New York....will they lose their accent?

I doubt it, if you are past your childhood, then it will be very hard to shake loose of your accent unless you really want to be rid of it. As well, if you are moving to, say, Manhattan, there really is no New York accent that is on that island. It is a mix of people; unlike say, if you moved to Brooklyn or Staten Island and were surrounded by people speaking well, how they speak.