These weren't the original actors, but very good look-alikes aided by some prosthetics. However the voices were of the original actors involved. So I'm willing to accept that this actually happened in orbit around Earth Prime-Time. As far as Toobworld Central is concerned, this is part of 'Star Trek' lore.
But here's the thing - the people working at the KFC are in the uniforms worn by KFC employees in the 1990's. (In the comments section for this YouTube clip, it was filmed either in 1991 to tie into "The Undiscovered Country" or in 1994.) And the equipment seen in that KFC restaurant is of that time period as well.
So it can't be a KFC from Earth Prime-Time in Kirk's era. I suppose it could be on a planet similar to Earth but which is only up to the 1960's style era, sort of like that Roman era planet in "Bread And Circuses" or the Nazi world, or the planet that emulated 1920's Chicago in "A Piece Of The Action".
As you can see, the idea was done to death in the original series. I think a better splainin is that the Enterprise was once again shunted back in Time, this time to the 1990's. They visited the Earth in the 1960's for the episodes "All Our Yesterdays" and "Assignment: Earth" and I would have preferred to have this happen just before the Enterprise left that time period after either adventure. But again, the uniforms are wrong - in the 1960's, the workers in KFC looked more like they were wearing old-fashioned ice cream parlor outfits of red and white striped shirts.
And as the Enterprise crew themselves are of the same age as they looked when the show was broadcast, this couldn't have happened during their time on Earth in the 1980's as seen in the movie "Star Trek IV: The Voyage Home". They were much older by then, plus they were wearing the latest in Starfleet fashion. On top of that, they were traveling, as Chekov would say, a Klingon "wessel".
So it would have to be after that time period, but before they actually went back to the 1980's. We only have available the denouement to an untelevised adventure for the crew of the starship Enterprise, which occurred at some point in the Toobworld timeline from 1991 to 1994.
As for the small differences in the looks of the crew? Something O'Bviously shorted out the time differential which caused the alterations to their looks. (This bit of wibbly-wobbly timey-wimey stuff was splained away in the "Children In Need" telethon special, "Time Crash".)