London rocks, but it is expensive.
I'd agree with the rent comments. They are pretty extortionate now as so few people are buying. So expect to pay £1400 PCM at the very, very least. We pay less, but we live in a scruffier area, so it comes with the territory.
Barnes is pricey, so you're more likely to get a bit more bang for your buck in Sheen, TBH. Is VERY family orientated there, and vey middle class, with lots of good schools. And lots of ex- pat American families, too. There are also little terraced houses to rent for minimum £1400 pcm, which is a bit nicer than a flat. Council tax is about £160 pcm.
Childcare in that part of London is around £1400 per month full time for under 2s. And if your kids are older, expect to pay £10 an hour for babysitting/school pick up.
Even though I don't drive, a car is useful. We use ours at the weekends, and it is very handy when carting all the child crap. So maybe join a car club if you do opt for no car?
I'd buy the winter clothes in the States, tbh, as they are most probably cheaper.