Martin,
i could not agree more about increasing the school budget at the expense of the defence one - though I think I would target the Trident programme, before the aircraft carriers... but I am afraid I just don't see that abolishing private schools would make any difference.
As I say, my belief is that those who send their kids to them now would continue to make the schools unwelcoming to anyone not from their background, and to subsidise them on top of the govt spending so they would continue to be the 'elite' establishments they now are.
I was brought up in Shamley Green, which is also one of the most expensive areas of the country to buy property in (think Eric Clapton and Richard Branson for starters!), and it too has a reasonable Council estate, but even so it was quite clear that the 'council' kids went to the local school and the kids of house owners generally didn't. If getting the kids into their preferred schools had meant moving to the relevant catchment area 99% of those people would easily have been able to, and would not have hesitated to do so. In this way they would swamp the catchment area, meaning that another school would be needed, and guess whose kids would end up going to each school?
The point is that, no matter how the govt legislates, this is a problem which is almost entirely intractable, since the people who most need to change their attitude (not something a govt can control incidentally) are precisely the ones with the greatest self-interest in NOT changing their attitude!
This is why I believe we have to tackle it from the other end and ensure that those from the 'less good' schools (which are actually often just as good if not better!) have positive discrimination in terms of universities, so that they can prove to their contemporaries from the 'elite' schools that they are just as good - and can get into the 'old-boy' network of University (as opposed to school) colleagues.
This does require a greater emphasis on schools and also (and I believe crucially) a re-setting of the expectations the kids learn to have of themselves as a result of the expectation the teachers (and  often also parents) have of them getting into 'good' universities. The poverty of expectation among so many kids is truely tragic!