Other mistake: Near the end of episode 5, Stevens is reading out previous numbers of slave units and B.O.S.S is telling him the new ones. For New York, the previous figure was 7203 and the new figure 7580. However, for Moscow the previous figure was 10003 (ten thousand and three), and the new figure is 110098 (one hundred and ten thousand and ninety-eight), which is a big difference.
Continuity mistake: (End of Ep. 3 and start of Ep. 4) When the Third Doctor fights the dark side of Omega's mind, the rings that the Doctor wears on the pinky fingers of both hands switch to his ring fingers, then back again to his pinky fingers, between shots. Then as Omega chokes the Doctor, the ring on his right pinky vanishes between shots.
Visible crew/equipment: (Part 3) After Omega has revealed himself to the Third Doctor, Jo, and Dr. Tyler, when he begins to speak privately to the Doctor the camera casts a shadow on Omega's body. Then after Omega turns around the boom mic dips into view at the top of the screen. And when Omega starts walking forward, the shadow of equipment (presumably the boom and mic) can be seen moving on the far wall, near the lower left of the screen. (00:03:10)