Rochester Sweeps had some hassles about this at one point - and since the blacking there is to do with being sweeps with sooty faces, changing colours wouldn't have really make sense, since soot only comes in black. As I understand it that's all blown over, since it's pretty clear there's nithing about their tradition which is anything to do with race.

Generally speaking Morris sides these days tend to use all kinds of colours when they do paint their faces, but I'm pretty sure that's to do with wanting to look looking more decorative, rather than worrying about people misunderstanding blacking-up as something to do with Minstrel Shows and all that stuff, rather than merely as disguise.

Since the theme of Mumming Plays, with St George and the Turkish Knight, is largely about reconciliation in the wake of conflict, I can't see them having any problems on that score. Traditional multiculturalism, in fact.