I've always felt that London is a separate micro climate and the factors there can often be completely different - I can see that coming across from some other people's reasoning.
We have 2 excellent local state schools, both of which are faith schools and we would not get into in a squillion years. Unfortunately as a knock on effect the rest vary between below average and fairly dire.
My DCs are at a not particularly flashy, non-selective private school which is only about a 10 mins walk away - its funny because for us we get the "community" thing from going to this school, as lots of children in the surrounding roads go there and they have a great network of local friends, more so than if they went to one of the state schools.
There is also a timing thing, that if you get a place at one of the private schools you have to be happy to forfeit the deposit and a terms fees if you want subsequently to accept a state school place as you hear about these comparatively late in the process.
Ability to get into good secondary schools is also a factor. "Good" round our way is definitely private.
School is very mixed by ethnicity but not much socially - we are aware of this and will try and get them doing more clubs etc in the wider local community to counter-balance.
I was fully state school educated and went to Oxford from a reasonably average comprehensive, so am not biased against it either way. But the options available to us in our part of London really didn't stack up. Moving not an option.
Cost is not an issue for us, but it is not a "flash" thing, more what we think is the riight decision in the circumstances.
Very happy with the individual school.