Plot hole: In episode 4, watch what Romana does to Professor Kerensky's time machine in order to make it work. It is surprising that the time machine works at all, considering she is supposed to be in Paris at the time. Romana wires up the time machine to a British (i.e. 3-pin 240 volts) electric plug instead of a "continental" 3-pin 110 volt plug.
Plot hole: The sketch of Romana is different when it's seen outside the café from the one seen inside the café (and just who's doing the sketch, and why?).
Plot hole: The Atlantic Ocean didn't exist in Early Devonian times (c.400 million years ago), so Scaroth's ship could not have been where the Doctor claimed. The Atlantic was formed when Europe and Africa separated from North America around 160 million years ago and is still growing. What is now the sea-bed was once covered up by several miles of Continental Shelf. The Early Devonian landscape would have been far from barren as plantlife was well established by then. So, either the Doctor's theory that Scaroth's exploding ship caused the creation of life on Earth is wrong, or it was much earlier than he said.
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 ★