Other mistake: At the end of the episode, the evil Shan is texting with (most likely) Moriarity. She is sitting at a desk facing a wall and yet the laser targeting ray appears directly on her forehead, which could only come through a window directly opposite her.
Revealing mistake: The banker shoots himself with a Sig 226R. When you see it lying on the bed it has a screw missing from the grip. In the first episode, Watson has a gun in his desk drawer. It is a Sig 226R also with a screw missing from the grip. Obviously they only have one prop gun.
Revealing mistake: When you see Sherlock and Watson in the museum, Sherlock leaves and nearly gets shot. You see him dive behind the large statue, which moves. That's unusual for a large statue, but looking closer, you can see there are wheels attached to the base.