Continuity mistake: In close-up, Illya is covering the phone receiver with his left hand, but drops the hand to his side when he tells the operator, "Never mind." There's an immediate cut to a wider angle, and his left hand is suddenly back on the receiver in the same position as before.
Factual error: The intro screen says we're in Hong Kong as a ship with Illya trapped aboard steams out of the harbor. We then cut to an exterior shot of the ship departing what we've just been told is Hong Kong - with, unmistakably, the Golden Gate Bridge in the background.
Deliberate mistake: In scenes at U.N.C.L.E. headquarters, the badge on Napoleon's lapel has the number 11 on it instead of the number 2, implying that Napoleon is in the wrong section. This is actually because during filming, Robert Vaughn believed that the number was "2" in Roman numerals (e.g. "II").
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Continuity mistake: When Napoleon, suffering from amnesia, finds his U.N.C.L.E. ID card in his jacket pocket, he opens it with one hand, with his other hand on the jacket's coat hanger. In the close-up of the ID card, he's holding it with both hands, then in the following wide shot, he's holding it with one hand again. The wallet also opens horizontally, then vertically, then horizontally again.
Continuity mistake: When Napoleon is chained to the table in the train room at the Thrush school, the shackle on the table moves from the bottom of the table closest to Napoleon to the top near the train switches, then back to the bottom near Napoleon.
Continuity mistake: When Illya talks with Carl Voegler in the woods, Voegler rests his bow on the ground, gripping it with his right hand and his left arm resting on top. In the next shot, he's gripping the bow with his left hand and his right hand is instantly down by his side.