At the beginning of the film, when Nick and his manager are on the train, Nick states that they're approaching the East German checkpoint. It's never stated when the film is set, but it must be before the end of WW2 (1945) if Nazis and the French resistance are still around, therefore East Germany didn't exist yet, as it wasn't established until 1949.
Agent Cedric (Omar Sharif) is driven to a wrecker's yard in an old London Taxi, which is then picked up and then dropped into a car crusher. A few shots later Hillary (Lucy Gutteridge) arrives at the dinner dance, in what is clearly the exact same car.
When Dr. Flammond tells Nick that the weapon has to be completed by Sunday, indicating September 24 on a calendar, Nick replies "That's Simchat Torah!" Simchat Torah is a festive Jewish holiday in which the liturgy includes reading the last portion of Deuterotomy, final book of Moses, and then beginning the cycle again with the book of Genesis. In 1984 Simchat Torah fell on October 18. Further, September 24 hadn't fallen on a Sunday since 1978 and wouldn't again until 1989, and neither of those dates coincide with Simchat Torah.
In the ceremony welcoming Nick Rivers to East Germany, the speaker refers to the country as "Ost Deutschland," which is the literal translation of "East Germany." East Germany was never referred to as such within the country; it was always called "Deutsche Demokratische Republik," which translates to "German Democratic Republic." (DDR/GDR)
When Nick enters Dr. Flammond's cell and Dr. Flammond is seen emptying spoonfuls of dirt from under his cot, you can clearly see him scoop dirt from a different pile underneath the cot rather than the tunnel he was nearly completing.
Chocolate Mousse (Eddie Tagoe) wears crossed belts of linked machine gun ammo, but all the cartridges are blank cartridges. This could be an intentional "mistake" since the difference between blanks and real ammo is so obvious.