St John's Wood is nice - good plan to look at places on the Jubilee line.
Clean air - I suppose all things are relative, and I suspect St John's Wood, especially with the parks, could be wonderful coming from HK.
Area 'not too congested with buses etc' - hmm, trickier - most people want buses as they make it easy to get places! I guess you mean you don't want to live on an actual busy road - there are lots of quiet streets around St John's Wood.
If you're going the private route for schools and are prepared to travel a bit you'll be fine just about anywhere in London - as always, money increases your options. There are international schools as well if you'd rather they followed a US curriculum if you're planning on returning to NY.
If you're planning on state schooling, do some research if you're planning to stay in the area you move to - in the UK children start school the September before they turn 5 so it may be a concern earlier than you'd thought (1st grade is the year they turn 7 if I remember correctly).
Nannies can get 'poached' if employers bring them with them from abroad but don't amend their salaries / terms and conditions to meet the increased costs of living and 'acceptable' nanny T&Cs in the UK - once nannies find out what they could be making on the open market I can see why many of them leave employers who are underpaying them.
If you have a nanny you know and trust who'd be happy to come with you it's worth considering but look carefully at her visa requirements.
London, and indeed much of the UK, can be pretty grey, but most of us don't know any different so can't really say if it's good or bad. The the advantage is that it's rarely too hot or too cold to go out with children, you don't need aircon and the humidity is generally bearable.
(NB if this is a work-led relocation the company will often have relocation consultants to help you through these kind of details - worth checking.)