Revealing mistake: When a Cyberman's arm is sliced off by the Robot's shuriken, you can see his real arm bulging out at the right side of his body. (01:00:25)
Other mistake: The Cyber Leader is supposed to follow the Master's footsteps exactly in order to get across the chessboard. He doesn't. This cannot be classed as a character mistake because the Cyber Leader knew what was involved. I believe this mistake was caused by the actor not being able to see the chessboard through his helmet.
Continuity mistake: After the Raston Warrior Robot has killed all the Cybermen, watch the dead Cyberman in the back right of the picture. His arm position changes as the robot teleports away.
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 ★