About 50% of the middle-class similar background children we knew at playgroup stage, went to state schools, the rest went private. Reasons varied: not believing in private education; passing the state 11+; meeting religious criteria; etc. Other than having more than one or two children making paying fees impossible, money was not usually the issue, indeed in a couple of cases Grandparent offers to pay were refused.
I think Dragon is right. London professionals work alongside lots of Oxbridge graduates and will encourage their children to aim high and to "have a go".
I also do not see any evidence that paying for education is buying Oxbridge places. If anything slightly the reverse. Two kids from similar backgrounds turn up to be interviewed for one place, one clearly polished and prepared , the other less so. The interviewer might well prefer to pick the "rough diamond" who has interesting and original things to say, and who could go on to have more to offer. In any case my observation has been roughly equal Oxbridge acceptance rates between the two groups.
Where things may differ is for Oxbridge fall backs. There seems to be an assumption that it is Oxbridge and "the rest" perhaps partly due to the extensive outreach opportunities for state students at Oxbridge. Private school kids seem better prepared through the process, in GCSE and A level choices and then in terms of where to focus applications, not just Oxbridge but elsewhere. So you may get more applying to, say, four top ranked courses with the expectation that you take a gap year if you don't get the place then get the grades. Easier for London kids as London Universities are a viable second choice. For whatever reason the focus always seems to on state school percentages at Oxbridge. Durham/Bristol/UCL/LSE/Imperial/St Andrews also struggle.
I also agree that the early deadline may be an issue. Easy at a private school geared up to having PS' written over the summer, and all UCAS forms submitted in October. (So staff can go on to work on the much more complex US Early Decision applications.) Less so somewhere where you have to catch staff at the start of term and ensure they move things forward promptly.