Revealing mistake: When Noah bangs his green arm down, in a close-up you can tell that the arm has green bubble wrap around it. (00:02:30)
Continuity mistake: When the Doctor and Harry arrive in the cryogenic chamber, they walk around for some time inspecting the chamber and talking. Towards the end, they notice a green slime trail newly appeared on the floor. But the trail was visible in the very first overhead shot of the chamber.
Revealing mistake: When Vira emerges from the hibernation chamber, the Styrofoam that forms the pallet squeaks.
Revealing mistake: As the cast move through the transom of the space station, observe the starscape through the lower windows. At certain points you can see a thin strip of light where the black curtain used for the stellar backdrop meets the studio floor.
Deliberate mistake: In episode three, the Doctor and Vira meet a half-changed Noah. He holds a gun on them, then drops it at Vira's feet and the door is suddenly closed. This is because the scene was originally longer and involved Noah begging Vira to kill him before the Doctor shuts the door. It was trimmed down to get it past BBC censors.
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★