Okay, the really schmanzy primary schools (those that do the whole blazer-and-dinky-little-boater thing that seems to be so important to some people) are: St George's Hannover Square (W1), St Peter's Eaton Square (SW1) and Our Lady of Victories (Kensington, but Cafflics only). I expect there are others, but those are the ones I know of.
There are lots of schools that are also v. good but don't do the fancy trappings. Whereabouts are you, Earlybird? Being churchy helps, but not essential. Check out the Good Schools Guide, they specialise in private schools, but also list the kind of state schools that people who can't afford private might be prepared to consider.
Apart from that, if you want a decent primary there's not much substitute for your own legwork, both virtual and actual. Ofsted is a good start, but there are some excellent schools you might not want to send your children to if you want them to have at least some friends from similar backgrounds to their own. If you look at a lot of Ofsted reports you can learn to read between the lines.
And there are some outstanding secondary schools as well. But you have to read the writing on the wall early and well to make sure that you meet the requirements, in terms of location, church-going habits or your child's propensity to pass assessment tests.
No-one said it was straightforward. But it can be done. And judging by the stress-levels of private-school parents of my acquaintance, that isn't a walk in the park either.
Expat: I'll see you there.