You could theme it more as a Torchwood type thing. A team of slightly normal people who have met one of the Doctors in the past but deal with Doctor Who type problems.
Several of the current batch of companions have been doing things when they met the doctor. Or re met the doctor.
The Doctor could be a visiting NPC, who pops in from time to time or is that last desperate help for when you are about to get the planet blown up.
Unit, Torchwood, Captain Jack on his own, the Alternate Mickey (the time travelling commando type from the alternate verse)or the Sarah Jane Adventures (may she rest in peace).
No reason why a Doctor Who adventure must be locked into following the Doctor round all the time, there seem to be alien visitors all over the place, enough star ships to hitch hike across the universe, other people who can time travel etc.
It could be that the new players starting adventure is when they are caught up in an adventure with the Doctor, they meet him and each other and have a crazy adventure through space and time. Then they are returned home.
Thing is now they know there is more out there it’s as if the veil has been pulled aside, now they can see what other people ignore or rationalise away. Several episodes have had people who know what the doctor was in them. The players now know the truth, they know that shooting star is an alien ship coming down to land, they read the stories in the paper and they know that those monster attacks everyone laughs about could be true.
So do they sit at home and worry or do they go and investigate for themselves.
Doctor Who is an entire verse, a Doctor Who game can be far more than just a trip in the Tardis and following the Doctor around.