I think skin colour is a red herring tbh, esp in a private school. They will be 'the American kids'. When DS started in his school there were some black kids in his year but they were the 'brits', not black - it really wasn't an issue. Are you concerned that they will be bullied because of ther skin or because they are 'foreign'?
In our school we have kids from all over the world, and of all colours, and they are usually described by what language they speak, not colour. My DS is the token 'english' kid (even though he is 1/4 english, 1/4 scottish and 1/3 ME!).
In the nursery at work the kids backgrounds are even more diverse (one child is described as '1/8th...') and there has never been any issues around colour, race or religion.
I 'get' why you worry, but it won't be an issue unless you especially want other black children in the class especially, but in London privates, classes can change termly.
Maybe if you can look at staff and look for schools with a proportion of black or other 'ethnic' teaching staff. Our school has a few black teachers, and a few auzzies, french, polish, italian...
Really, don't worry about it. Just find a nice school - there are loads around there. Even in the shitty state school, out in the wops, I went to, one of the coolest girls was the only black (ie of african origin at some point in the family) and that wasn't to do with her skin colour.
BTW the state school one side of us is mainly eastern european, the other side is mainly arab kids. It's just the way it is where I live - very very few white brits, probably more black people actually!
If work is picking up the tab, they possibly will help you out with school suggestions, or pay for a trip over to find a home/school before you move. They may even pay for a search company to help you find something suitable.