I wish there was a Doctor Who crossover during Star Trek's original run. Patrick Troughton was the Doctor then...that would have been fantastic. But a dream is just a dream.
Yeah, that would've been pretty much impossible. British audiences never saw Star Trek
until July 1969, three weeks after the final Troughton episode ended and about six weeks after TOS's final episode aired. And US audiences didn't see Doctor Who
until the '70s, in limited release, and then more widely in the '80s. So at the time both shows were on the air, neither show's audience would've had any awareness of the other, and indeed neither show's producers would've been likely to have much awareness of the other.
I just don't think most franchises lend themselves to crossovers with Trek, particularly Dr Who. Not only do history and technology not fit, neither do the laws of physics. And as Scotty says, 'You cannae change the laws of physics...'
Yeah, as I remarked, that's the crux of the problem. The fundamental worldbuilding is often so different that you just can't reconcile different universes.
I think it's been suggested that it might be possible to fit Blake's 7
into the Trek continuity as a dystopian future, one where the Federation has turned oppressive and evil. It probably doesn't fit due to the lack of established Trek races, though. I do think it's possible to fit B7 into Doctor Who
's much looser future history, though.