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.
Thanks for the Doctor Who history (I did not know when it actually came over to the States).
I knew it was an unrealistic fantasy but it's still something I would have loved to see.
Of course in a lot of cases Crossovers are badly written in my opinion and don't often live up to the hype.