Perhaps they could wear black ski-masks or balaclavas.

That aside, I found this on John Kirkpatrick's website, taken from an old interview:

"We used to sing the words, "There was a little nigger but he grew no bigger ... " but we've changed them to avoid offending black people. I don't think there's any cause for offence in blacking up, but singing about niggers is rather different. I must say we have rarely come across blacks where we live, obviously by travelling around we see more, but they usually either seem to understand the point of the disguise or they don't make any connection at all. The first time we ever danced out, we were all driving up to the pub in our blacking for the first time, nobody could recognise anybody else, and this carload of Africans turned up in full flowing robes. This had been set up by one of our members and he hadn't told the rest of us, so we were shitting ourselves even more than we were already. But they loved it. "