Continuity: No one seems to know which daughter is which. Margo is named daughter number 2 in Act II's title, but the dialogue calls her daughter number 1. Act III's title says Imogen is number 3, while the dialogue refers to her as number 2.
Continuity: On the ground, the mini-jet has a large yellow U.N.C.L.E. logo on its nose. In the air, the logo keeps disappearing, and the plane's tail assembly changes configurations.
Continuity: Solo picks up a book, The History of Pacifism, and shows it to Illya, holding it at the bottom. Cut to a close-up insert and his fingers have instantly migrated up to the middle of the book. Cut back to the wider shot, and he's no longer holding the book at all. It's back on the table.
Factual error: The jailed student protestors are dancing in their cell to a jazzy tune. But they have no radio or record player - just one guy with a guitar. They seem to have conjured the accompanying saxophone, jazz clarinets and percussion section out of thin air.