Plot hole: The linchpin of the plot makes no sense whatsoever. In an effort to diffuse international tension, the superpowers would allow Britain to publish the codes that would allow anyone in the world to launch their nuclear missiles? And then, after going through the drama of not one but two countdowns, it's revealed that the superpowers can just activate safety over-rides to prevent the launch. So how in Hades did the SRS have any threat whatsoever to wield over them and issue their demands?
Revealing mistake: In the last part, the giant robot steps on a soldier. There are several things wrong with this scene. 1) You can see the matte halo around the robot's foot. 2) The robot does not leave a footprint. And 3), after the soldier gets squashed, his arm is visible waving over the robot's foot.
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.
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.
Continuity mistake: In Episode 2, when the Doctor calls Harry from the Control Room and Harry enters the room next to the Cryogenic Chamber, there is an oblong yellow panel next to the door. It extends from about Harry's waist line to about 200mm below the top of the door. But in Episode 3, when Noah calls Vira from the control room part way into his transformation, and everyone runs into the adjacent room, the panel is mounted differently. It now extends from about Harry's mid-torso, to the top of the door frame. (00:03:35 - 00:09:32)
Continuity mistake: When Sarah falls from the scaffolding, a plank of wood saves her from death. Trouble is, she fell a foot from the outside of the scaffolding and most of the plank was inside the scaffolding, therefore her feet would have hit it and she could only have landed with her head pointing inwards. But when we see her, her head is pointing outwards. (00:01:25)
Revealing mistake: During the Doctor's early conversation with the Time Lord, there is fog swirling around them. When the Doctor agrees to take on the mission, it suddenly appears very concentrated around the tip of the rocks behind them, as if generated from a source behind the rocks. Normal fog would not do this, and it's clear that dry ice was used as the fog effect (DVD commentary confirms this). (00:02:45)