If anything Augustus looks to be modern eastern european (from its famous statue).
What's not correct about "white" when depicting roman aristocracy? From all the statues we have they were certainly neither asians nor black nor from india nor descendants of, say, the aztecs.
Here's a reconstruction using AI of 30 roman emperors and I'd say 27 of them look totally white.
https://www.boredpanda.com/realistic-recreations-ancient-scu...
Heck, they even made Augustus with blond hairs and blue eyes (I have no idea if that's correct: all I can say his is marble statues looks like the modern easter european type to me).
Now I can understand they didn't sound british when speaking, but I don't see why you write "white, british actors". If anything, and there's nothing racist in there, "black british actors" would have been weird no!?
Or maybe you meant to say most of the aristocracy probably had the "mediterranean" type? (which I don't think is true and which, anyway, is classified under "white" I think?)
EDIT: TIL: Augustus' biographer, Suetonius, wrote that Augustus indeed had golden hair and clear eyes. This comes at a surprise to me.