Many of the schools mentioned seem to be secondary schools. If you are looking for Private preps and want to live in South Ken/Chelsea (both lovely areas though well out of the financial reach of most Brits) you might look at prep just south of the river, including Newton Prep.
You will then get a wider economic if not racial, diversity. Plus it can be assumed that parents who choose to live in inner city south London close to Brixton (Stockwell, Clapham, Vauxhall) are reasonably relaxed and open.
I think Thaliablogs is right about the smaller black middle class in London. However there is one and they use private schools. It will depend on the school but I hope I am right in thinking that the black and mixed race children we have known and their parents have not experienced any problems.
North of the river there will be huge diversity in that children at schools lile Glendower and Eaton Sq will come from all pver. Russian, American, Asian, Arab, European etc. Some will be visibly ethnic though only a small proportion of these will be black.
A choice really. Do you want your kids to be part of a rich international and cosmopolitan community reflecting the Chelsea/Souyh Ken population. Or something less international/more British (though still will quite a mix) greater ecomonic diversity, and probably a few more black or mixed race kids.
Either way things will be diverse and your children shoild not feel any more different than their peers.