Oxford and Cambridge both have something along the lines of 40-50% private school students. However that's not because they're discriminating, but that state school pupils are put off applying and so don't, hence the skewed ratio. Change is happenning, but slowly.
I'd imagine the same is true at other unis, too. Just because there's a high percentage of private-schooled people, it might well dissuade state-schooled pupils from applying, hence maintaining the ratio for no real reason.
So the moral is, apply where you like- even at what are percieved to be the most public school dominated universities in the country, you'll have at least 50% of people witha common educational background.
Besides which, it doesn't really matter overall. No-one ever feels the need to talk about it, or brag, or boast. The only time it has come up is in a joking, bantering kind of way, and you can well give as good as you get. Just because someone payed money for their education does not make them an intrinsically better person, nor academically superior- you've got just as much right to be at the same unis so apply where you feel comfortable, not due to some percieved need to stick with a large percentage of similarly schooled pupils.