

They could definitely figure out a way to make it work! It could be pretty fun, and I would at least have a big grin on my face when I first saw the TARDIS on the Enterprise bridge!
And yes, I’ve noticed that too. The world of Doctor Who wouldn’t be a great place to live compared to Star Trek, at least from what we’ve been shown. Even the tone of the episodes is often darker, there are quite a few classic and modern episodes that are just family friendly horror. Trek has done some of that over the years, but it seems Who leans into it more, or at least I notice it more!
I don’t think they usually worry too much about continuity. They’ll try to make things line up and explain inconsistencies sometimes, but there’s a lot of times the writers say “wibbly wobbly, timey wimey” and try to tell a fun story without being locked into things. Seeing that approach for so many years is probably why I normally don’t mind the inconsistencies between the various Star Trek shows!