The thing is we have 5 DCs, live in an area with either good girls schools, one good coed which we actually have a chance of getting into (but in reality wouldn't) and no good boys schools. The problem is though that we have four DSs!
Send the DSs down the coal mine to pay for private for your DD? 
We moved from London to Kent specifically for grammar schools. DC1 got in easily and is enjoying it and doing really well, so no regrets there.
I do miss London quite a lot personally though. And I worry for DC2 (much younger) who is much more outgoing and has interests that would really benefit from the social and cultural stimulation of London. The thing about London is that even if the area where you live has shortcomings, the population density and cultural intensity is such that you're not far from somewhere that can supply the things you don't have. It's a bit different when you have to make an expensive 2 hour train journey to get those things.
Schools are important though, and if you don't have the money to either (a) go private, or (b) buy a house in a good cathment area, then grammars are about the only other way of ensuring a worthwhile experience for academically able DCs.
With five DCs, I imagine (a) is likely impossible for you, and even (b) pretty hard. I know you said you didn't want to hothouse, but I would maybe consider buying somewhere in outer London that has access to grammars and some decent comps as well, plus reasonable transport links so that journeys in are doable for other things, and for when they get older. Then even if some of them don't end up at ideal schools, you've got quite a few bases covered for however they develop.
It does my head in trying to consider the talents and interests of both my DCs, and then work consideration for both DW and myself. It gets so that there's nowhere you can go that's OK for every consideration. I can't imagine what it would be like with five.