Lots of good points made here. And I am not talking about Xenia.
We're in central London. Most people I know with young kids have moved or are planning to in next few years. I can't imagine leaving - I love it. There is so much on offer for all of us. Our jobs are tied here anyway and commuting long distances is something I have no intention of doing - it adds way too much pressure on to the family IMHO.
My DCs (4, 1) will go to local 'good' (Ofsted) primary, which local MC-mummies roll their eyes at, since there are a plethora of pre-preps/preps and a couple of Outstanding schools with miniscule catchments, nearby). Has v high FSM, ave SEN, v high EAL. However, I do not believe that my children are going to catch something from children who have different backgrounds from them. I know kids locally who've been through these schools and gone on to fabulous things. I see no reason at all why my children can't do the same (jeez, if I can't have confidence in them, then who will?!).
I have cut back my hours (sorry Xenia - I am morally reprehensible) so that I can be around my children significantly more than I could have been otherwise, and this is part of what rules out school fees for now. However, given my Oxbridge experience, I know there were lots of people there (and at every RG uni too) who went to below par (even failing) schools - every kind of school actually, despite the hype. I also know plenty of wasters who went to very expensive private schools. A good friend left his expensive boarding school at 17, with 2 GCSEs and went on to drive trucks for a living. One of my best friends was inner London state educated all the way through and has a first degree and a PhD from Cambridge. In both cases, it was the involvement and interest of the parents, family situations and a whole host of things beyond the school gate which probably had the most significant impact on the outcome. Of course the schools are a factor, but they are just one influence. (I'm not anti private school by the way - just saying they are not the be-all and end-all).
I think that you generally have much better choices, school-wise, in London that you would in your average rural village.