There’s one pretty common complaint I hear fairly regularly about the possibility of playing in a Doctor Who campaign that, while I don’t necessarily agree with, I understand. I’ve often heard it said that his story is already told/unfolding. We know how the first 10 Doctors regenerated, so it’s hard to tell new stories without stepping on some creative toes.
If that’s something you and your players aren’t comfortable with, consider the Meddling Monk.
The Monk has an interesting place in Doctor Who lore. He was the first other Time Lord we ever met in the series, but unlike basically every other one we’d meet, he wasn’t really a bad guy, per se. Quite the opposite, in fact—he was interested in helping humanity achieving its true potential, even if it took a little timey-wimey to achieve it. We first meet him trying to stop the Norman invasion of England, so humanity would enter a new Renaissance.
In fact, it seems like his only crime was meeting the Doctor at the wrong time—the First Doctor had a policy of “no changing history whatsoever,” so the Monk was cast as evil, but I don’t know if Ten or Eleven would’ve had the same violent reaction to him. And so they clashed a few times (mostly out of the Monk’s attempts to get revenge), the Doctor easily won, and last we saw, the Monk was stranded on an Ice Planet. Which makes the perfect starting place for a campaign!
So if your group doesn’t want to re-tell the Doctor’s story, but (like me) doesn’t like the contrivance of “oh yeah, I ALSO ran from the Time War! What’re the Odds!”, consider the Monk. Like the Doctor, he loves humanity, wants to see it thrive, and is more of a schemer than a fighter. Unlike the Doctor, though, he’s a little more morally flexible, and—this is important—on his first regeneration, and would have no reason to show up for the Time War, stranded as he was.