Continuity mistake: Mark, April and the professor leap from a boat seconds before a bomb goes off aboard. But in the shot of the boat exploding right after they jump, there's no one in the perfectly calm water, and the empty rubber raft is still alongside the burning boat. Somehow, though, the escapees are rowing to safety in that same raft a few shots later.
Continuity mistake: The bad-guy bellman knocks Mark out, then zips him face down into a garment bag. He then places the bag, without turning it, onto his luggage cart, and the outline of the body inside (a dummy, judging by how effortlessly he lifts it), is now facing up.
Continuity mistake: Merek's bird is sitting on his shoulder facing backward. In the very next shot, as he stands, the bird is suddenly facing forward. When he starts to sit down, the bird is still forward, but in the next shot, a close-up of him sitting, birdie is facing to the rear again.
Continuity mistake: Mark injects April with plain water, telling her to play dead. She shouldn't have to play. Water in the blood stream can create oxygen bubbles, which can be lethal in a matter of minutes. He should have used saline or glucose instead. Those solutions have been de-oxygenated and purified to prevent giving patients the potentially lethal "bends."
Continuity mistake: Highly-trained secret agents are supposed to know their own IDs. When April reports to HQ over her communicator, she identifies herself as "April Dancer, Section One." She works, as do all the U.N.C.L.E. field agents, for Section Two.