Bonkers I have no idea about the overall numbers of bursaries in private schools. Apart from anything else you would be including chalk and cheese, since there are some private schools out there who exist entirely to make money for their owners / share holders, . I don't know for instance why anyone would be paying for their children to attend one of the schools run by GEMS who are very open about their mission to make as much money as possible out of the UK market and are lobbying the government to be allowed to make money out of their participation in the Free School programme (hopefully the D of E have realised or will realise there is little reason to send money their way as well).
However at most of the top schools there really is a commitment to have as diverse a student body as possible. As far as bursaries go this would be a representative example.
"Over 2012/13, some 131 pupils (15.2% of total number of pupils) received financial disbursements, at a cost of £776,100 (6.1% of gross fee income). Of these, 42 receive financial support through bursaries, of which 21 were full bursaries. For 2013/14, bursarial support has been increased to £936,765." There is a bursary appeal aimed at increasing that pot. That would be in addition to the cost of involvement by pupils and staff in activities in local state schools, sharing initiatives such as the Model United Nations with local state schools, there is a formal partnership between local state and private schools for that purpose, mentoring schemes for state school pupils and offering the sports etc facilities to local state schools for free. I don't think this is an unrepresentative example, indeed it isn't one of the schools that is particularly wealthy compared to a school like Eton, nor is it a direct grant grammar where ex pupils who were given a chance of social mobility tend to be more generous in terms of providing the funds for future pupils to get the same sort of chances.
My DDs school is far more diverse ethnically than the state school in the area because pupils travel to it from a very wide catchment. A lot of immigrant parents for instance are willing to stretch themselves to fund their child's education and there are significant numbers of Asian and Eastern European pupils. In my own DDs friendship group her friends are from families from Sierra Leone, Tamil Sri Lanka, Kenyan and Ugandan Asian, Poland and Singapore Chinese.