Welcome to NY!
Just to quickly clear up W2BS's comment, I'm not a realtor. I was working on some marketing. I do live in Westchester though! My eldest DC was the same age as yours when we arrived.
Basically, there is a reason lots of expats can be found in the same Westchester towns, it's because they have very highly regarded schools that are used to international students transitioning for 3 years.
Scarsdale is one such town. Personally, it's too flashy for me and I prefer a bit more distance between the houses. But you can't knock its school district, regardless of how competitive it is.
I would look at Mamaroneck, Rye, Larchmont, Irvington, Pleasantville and Chappaqua. I wouldn't go further north than Chappaqua as then the commute gets long, although there are plenty of Brits up there. Armonk is nice too, it just doesn't have a train station. You need to get the train from N White Plains so you would have a drive first. It's a very pretty drive though!
I love Tarrytown but a lot of Westchesterites sneer at its school district, mainly as it has a relatively high Hispanic community. The schools there are good just not the best of the best. It's a vibrant town with a lot going on. I'd also maybe look at Croton.
It really depends what you want. Chappaqua has fantastic schools but it's dead by 8pm and you need to drive everywhere. It's right next door to Mount Kisco though, so there's a lot more going on there. Pleasantville is lovely and lots of the homes are walkable to downtown. Schools are very good. They were one of the first to opt out of state testing as they felt it was to the detriment of the children's learning experience. It also has an art house cinema, plenty of restaurants, etc. I love Pleasantville.
Irvington is quite cool too. Loads of excellent places to eat. The housing stock is quite varied. Croton is quite arty and more laid back.
As I said, it really depends what you're looking for. Feel free to pm me if you want more specific information. Happy to help in any way!
And, as an aside about NJ, it's no good for the OP but parts of NJ are lovely. For those in the financial district, Bergen County has some lovely towns that easily rival anything Westchester has to offer. It's really not all slums, Jersey Shore or RHONJ!