The whole series is about people screwing around with time travel, so there's always an easy explanation for why things aren't always going to line up perfectly.
Star Trek and Doctor Who are not changing canon, they are simply adding to it. All of the stuff in the earlier seasons and series still happened, the new shows have just been adding more to it.
I read up a little on the Timeless Child, and while I'm not a huge fan of it, it just seems to add a whole lot of unnecessary complications to the Doctor's backstory, and I'm not a big fan of making them some extra special character even before all of the journeys as the Doctor, but I really don't see where it's ruined the character or the show. All it did was add an new backstory to the Doctor, but it didn't destroy anything in the process because everything we've seen and learned about the Doctor's life before the show, and everything we do know about still happened, we just now know that some other stuff happened to them before that. Hartnell and Bradley is still there as the First Doctor, or at least The First Doctor that we met, he just wasn't the first life that this character had. They already introduced the idea of there being other lives that didn't count with The War Doctor, and there might have been a few in Classic Who too.
And it's also the kind of thing is easy to ignore by writers since it all happened at least thousands of years ago, and going by what the TARDIS Wiki says, there's an easy out if someone really wanted to just retcon it away. Going by what the Wiki say, since the whole thing was something The Master stumbled across in the Matrix, you could always just say that he found a way to hack into the Matrix and fake the whole thing, and any references to it outside of the Matrix were all set up by him somehow.
And the same is true of the new Trek shows too, there is nothing that they have introduced that has changed canon, all they've done is add new layers to it. Yes Michael Burnham is a big addition to Spock's backstory, but we never really learned that much about his family or backstory anyways, so it's easy to just stick her right in there.