There used to be loads in North London, but most of them (Hampstead, Haverstock, Highgate Wood, APS) have succumbed to the uniform juggernaut over the past few years.
The ones still holding out (and likely to continue, fingers crossed) are Fortismere, Camden Girls, Parliament Hill and Acland Burghley. They're all state schools with varying levels of sought-after-ness, but crucially all have a critical mass of middle-class parents who feel very strongly about the liberal ethos, so the non-uniform aspect is very much part of their identity.
Most of the high-profile non-uniform schools outside London have a similar intake, eg Cherwell in Oxford, and formerly KEVICCS in Totnes (although they are going the uniform route too).
Private schools with no uniform tend to be either alternative (eg. King Alfred's) or so sought-after that the absence of uniform adds to the mystique (St Paul's Girls).
Interesting article on non-uniform schools here.
I really cannot see the logic of assuming a school will magically be transformed by making all dc wear polyester blazers. I dont find the 'preparing pupils for work' argument convincing either - the only work that has that kind of dress code is eg. working behind the counter in a building society. And the same goes for 'high academic standards' argument - have the people making these pronouncements seen what university students and lecturers wear? Would standards at Oxford, Cambridge and Imperial rise further if everyone was made to wear cheap suits?
I'll stop ranting. 