I live in Caterham and my children are at Trinity School, which is part of the Whitgift foundation with Whitgift school. My children had places at all three. They have friends at both Whitgift and Caterham. We chose Trinity to a large extent because it, at the time, was by far the most ethnically diverse and still is. Whitgift also was and is, but maybe not quite as much (and a very different feel). Caterham was a very, very white bubble imo, outside of the Chinese and Russian boarders that they always reference. Having said that, I visited recently with my youngest, and I was struck by how much the ethnic mix had improved and it was visibly less uniformly white. Still nowhere near as ethnically diverse as Whitgift and Trinity, but definitely a better balance than when I'd last been there 3/4 years ago. Whitgift is a boys school of course and Caterham co-ed which might be important for you. Trinity is a gain a nice compromise for me as girls join in the sixth form.
Caterham itself is far less diverse than Croydon, of course, despite only being 20 minutes away, and that reflects the London/Surrey demographics really. That's also improving, gradually, I think.
I don't know any of the other schools personally, but Emmanuel strikes me as also very ethnically diverse, like all the South London independent schools really. Kids travel from miles to get to all of them, as well as local people, so the pp is incorrect imo that there is a direct correlation with the ethnic diversity of the area, but South London - more diverse than Surrey will always be true I think.