Private school students typically score around half a standard deviation (7.5 points) higher on IQ tests than state school students, at the age of 17-18. Correspondingly, they do better, on average, in their GCSEs and A-Levels, and almost certainly do better in the aptitude and admissions tests deployed by Oxbridge as well as in their rigorous interview process. Thus, private school students are
bound to be overrepresented at Oxbridge.
It's likely that high-achieving state school students - and high-achieving students from disadvantaged backgrounds - are less likely to apply to Oxbridge than high-achieving private school students. Yet, Oxbridge already carries out extensive outreach to those from disadvantaged backgrounds to encourage them to apply.
The BBC article states that "high-achieving independent school pupils were twice as likely as state school pupils to apply to Oxford and Cambridge, even with the same ability and predicted grades", yet I couldn't find anything as explicit as this in the report. The report did, however, state that "just 25% of state school students with grades A*A*A and above apply to Oxford, compared to 37% of such students from private schools." Of course, part of this difference could be due to state school students not believing that they have the right GCSE grades to apply to Oxford, but it seems likely that a large proportion is due to different attitudes to Oxbridge among state and private school students.
Oxbridge also take into account contexual data in admissions. They could go further and explicitly lower entry requirements for those from disadvantaged backgrounds, but they would first have to study - and take into account already existing findings on - the degree to which academic achievement is down to coming from a "privileged background". The fact is that genetic differences play a far greater role in explaining differences in academic achievement, and IQ, than "nurture" does. Nurture, including the home and school environment,
explains at most one-third of the variance in GCSE scores, and
much less (around 16%) of the variance in the IQ scores of 17-year-olds.
Thus, even if we were to get rid of all of the environmental privileges that private school students enjoy, completely equalising the home and school environments,
and encourage the same proportion of state school students to apply to Oxbridge as private school students, we'd still see overrepresentation of private school students, due to differences in average innate ability between the two groups.