Everyone seems to be ignoring YoDiggity's excellent post on page 9.
The educational divide between the poorest and the richest in our society is massive and unimaginably complex. Banning private schools would be a political knee-jerk sop to the lower-middle classes which would achieve, ultimately, bugger all.
The problems do not necessarily lie with the schools, how good they are, how much money they have or what facilities they have (or lack) access to. There is also a problem with the social makeup of the pupils and their backgrounds. You could take a child from the poorest background and put them in the best school in the country, but it will mean nothing unless that child has had the value of education and the respect for it instilled in him, and it is this ethos that is missing in the poorer parts of the community. As Yo said, there is a reason schools in expensive leafy suburbs are generally good, and why those in poorer, inner city areas are generally worse.
Now this is not to say that those from low socio-economic backgrounds are thick or lazy - they merely have no incentive because you have generational lack of rewards, motivations and opportunities. If their parents are uneducated or have a palpable lack of respect for education (often lumped in with "authority") then the kids will pick up on that and simply won't care about school. It is easier to stay in your comfort zone of what you know rather than push for something that noone else you know has ever done, something that seems so out of reach you may as well try to go to the moon.
My brother and I were the first generation on my mothers side to go to university (dad was the first ever in his family to go to uni), and the first ones ever in both sides of our families to be privately educated. My mum's family is solid Yorkshire mining stock, my dad's family is slightly higher on the ladder, though not by much - nurses and army navvies. I see some of my cousins, of an age with me and my brother, on my mums side suffer from this same "don't care" attitude, which they are now passing onto their children. And so the cycle perpetuates. I see some of my cousins who have had a good ethos instilled into them from a good age, and they're now in fairly solid white-collar, middle-class jobs. Our side of the family is openly mocked by the ones who aren't doing so well because we've broken ranks and now we're "posh" (i.e. educated - I can confirm we are neither rolling in cash nor posh, unless having an old Fiesta and owning a labrador makes you posh?). Imagine a child trying to break out of that quagmire against their parents and siblings judgement?
There IS a way out, but it will not be found by banning private education. There needs to be a huge cultural shift in attitude to get people to use the opportunities that they have.