SOrry, I am new here and didnt have time to read all the replies so I apologize if I'm repeating.
I was born in Brooklyn, we moved to Westchester when I was about 10 and my parents still live there (I lived in Moscow, then Manchester, dh is from London, we dont know where we're going to be in a month ha ha ha)
Anyway,It sounds like some areas of Brooklyn would suit you best. Frankly I think that Westchester, and most of New Jersey (someone mentioned Hoboken) would be a real shock to you coming from s London (which by the way I love - we want to live in Stockwell). Westchester is extremely well-to-do (almost no rental market, BTW) and very white, also quite conservative. It certainly was a huge shock to me when we moved, and I continue to feel uncomfortable with the people and place for the most part.
Brooklyn (depending which area - it's quite big) offers what you're looking for, I think. For example, Carroll Gardens would be about a 45 min commute to upper Manhattan, and is a lovely, very child-friendly area of brownstones, lots of parks, independent neighborhood shops where the baker knows your name, etc. Rents have risen, but are still not outrageous, and it's famous for it's relatively large gardens (if you get a ground-floor flat). Equally nice would be Park Slope - I think a bhit hipper, more restaurants etc but next to Prospect Park which is Bklyn's central park, still definitely a neighborhood feel and a reasonable commute to the city. If you really have some cash, you might also like Brooklyn Heights -obscenely gorgeous views of the harbour and Manhattan, a quicker commute into the city, one of the nicest residential neighborhoods in all of NYC imho.
As someone has said, if you want to actually live in Manhattan, it has to be Upper West side. Although the rents will deffo be higher than in Bklyn. And although the area is nice, I personally think it is less of a neighborhood feel (bigger streets, more traffic, all around busier and more hectic). What I love about Brooklyn is you have all the benefits of the big city, but still a quite neighborhood to come home to.
I hope this helps, and enjoy your ny adventure!