Continuity mistake: In the scene where Dr. Dawson and Olivia are seeking Basil's help at his house, Basil sits down and starts playing the violin, holding it in his right hand and having the bow in his left. When Basil the stands up in another shot, he is holding the violin in his left hand (like it's supposed to be) and the bow in his left.
Revealing mistake: When Ratigan is sneaking up on Basil inside Big Ben, there is a shot where both characters have their backs to the camera. In this shot, there is a piece of equipment directly in front of the turning gear they are standing on. This piece of metal can be seen to flicker for a few moments.
Continuity mistake: During the scene where Professor Ratigan and Mr. Flavasham are in the dungeon, the robot pours the tea into the cup. During the next shot it completely vanishes. Then a couple of minutes later, the robot puts sugar into the cup where the tea was. The tea is still gone but it makes a sloshing sound.