Revealing mistake: When Steed comes out from hiding behind the spiral staircase and wall, the rim of his hat touches the wall, which wobbles, revealing a fake wall. It's very noticeable, because there is a shadow cast from the staircase to the wall where you can see it moving.
Continuity mistake: When Steed, Mrs Peel and their travelling companion Mr Smallwood arrive at the pub, it is pitch black. However, when Smallwood leaves a few minutes later to go and visit his brother, it is fairly light again. All subsequent shots of him outside continue to be well lit, whilst in the shots looking out from the pub it is dark.
Continuity mistake: When the butler shows up at Steed's apartment to kidnap Tara, they fight and her culottes are unfolded at the knee. After he has knocked her out and put the last jigsaw pieces in the puzzle, he goes to pick her up to carry her out, and her culottes have been folded up at the knees with no way for that to have happened.
Plot hole: The bad guy somehow knows what is going to happen (via the sketches) even though the events haven't happened yet, like knowing Mrs Peel was going to show up at Pool's mansion in that catsuit and hat, and that the scenes were going to play out like the sketches.