On areas, the key in London is the transport system. You won't be driving into Oxford Circus.
Get a tube map and look at where it's easy to commute to.
So, there's the Victoria Line, Central Line, and Bakerloo lines going straight to Oxford Circus.
Tottenham Court Rd and Bond Street are really close too, so you've got the Northern Line and Jubilee line to look at too.
That's a lot of places to get started with.
Can you get anything in Hampstead or Highgate in your budget? I'd love to live there. Beautiful houses, the huge Hampstead Heath right on your doorstep, great independent shops, loads of place to go out, easy access to Camden Town if you want hustle and bustle. Tubes into the centre when you want to be there.
My info on schools is about 25 years out of date though!
Having grown up in London, I'm surprised at some of the places people are mentioning, they're really far out, not what I'd consider the London experience if that's what you're after? But if what you're after is a bigger house in the burbs then maybe that's right for you?