Plot hole: In order to build up suspense, Virgil and Brains stand on the Crab Logger as its dangerous fuel is pumped out, despite the fact that it is hanging on the edge of a precipice. In fact they could have left as soon as they had plumbed in the pipes to empty the fuel tanks - the whole procedure is controlled by Scott in Thunderbird 1 and they make absolutely no contribution to the operation at all. They do not need to disconnect the pipes; they are left behind and dropped from Thunderbird 1 as planned. They just stand there chatting about how much danger they are in until jumping clear at the very last second. Silly buggers.
Plot hole: The scientist yells for Igor to capture the escaping Castaways, to which Igor responds "capture them yourself" in Ginger's voice. The scientist appears surprised to find out Igor is not Igor, but how can this be? He had to be the one to pull the lever to switch everyone's personalities and bodies around, so he should have already known.
Factual error: U.S. marine utility uniforms have the Marine Corp Emblem and USMC printed in black ink on the left shirt pocket. Private Duke Slater's uniform does not have it printed on his left pocket. His uniform also doesn't have the proper pocket flaps. (00:07:00)
Plot hole: Solo blows open the ceiling vent of Illya and Marion's cell, and she stands on Illya's shoulders until Solo can reach her and pull her out. This leaves Illya with nothing/no one to stand on, no way to reach the vent and no apparent way out of the cell. But he's out just the same in the next scene, with no explanation as to how. (00:42:40)
Factual error: The sea plane that lands to rescue Savage at the end is a type not built until after the war. And it's marked "U.S. Air Force," an entity that also didn't exist until after the war. During WWII, what later became the Air Force was known as the U.S. Army Air Corps.
Continuity mistake: When Drake is leaning against the back of the park bench while talking to the tailor, in the shots over his shoulder, his watch is partially covered by his shirt cuff. In the shots from in front, his watch is completely exposed, despite his arm never moving. (00:12:25 - 00:13:00)
Continuity mistake: Hap puts his hand around Bud's side. Camera angle switches, his arm is on Bud's shoulder. Another switch his hand is no longer on Bud. Camera angle switches once again and Hap's hand is being removed from Bud's shoulder. (00:16:39 - 00:17:00)