There's not to many british tv/movie actors/actresses, so there will be plenty of spill over into other productions, even if its small parts. The Saint, Avengers, etic. Belive Mis Drake was in Star Trek ( Maybe? there is a photo of her on the transporter pad, but could be a fan made)
Desilu did this to, Watching Mission Impossible old series, it is replete with star trek stars, sometimes doing 2 or 3 different roles, like Mark Leonard.
Up to Episode 21, Still pretty good.
For a Reboot, Could take the premise of "10 years in the future, 1980" and say make it 2030 or 2035 to explain away some advances in technology. So Start off "Today" with Straker getting approval to form SHADO after confirming UFO's exist and that there hostile.
So take the 10 years from "Now" and have Shado set up a moon base, observation satellites, and Sky Divers. Have it a UN agency, so the General Straker goes against is a UN official that occasional hostile, maybe have an AI component, with him saying computers and drones could do a better job. Which with the original series, there was some hostility and infighting between them.
The show is a bit like Godzilla, yes the UFO's and aliens are there, but not every episode deals with them, and alot of the drama is Earth/moon based.