Continuity mistake: Hastings has just met the Ingelthorps, and now is outside so John Cavendish can introduce his wife. In the wide shot it's evident that John lowers his arm, but in the closer view his arm is still held up high before Mary speaks. (00:08:10)
Revealing mistake: Japp is waiting for Van Braks to make his move, and when he is hopping on a taxi for the airport, he begins to tail him. When the car moves, reflected in the door is a modern No Parking cone in bright yellow (incidentally, they also pass by a parked car with a FUU 692 plate that is a bit too recent for the series, being an Austin Big 7, first produced in 1937). (00:41:05)
Factual error: Assuming the episode is in line with the others of the season and it is set in 1935, a few vehicles shown are slightly too recent. The plane waiting for South Africa is a Beechcraft Model 18 (first produced in 1937), the train going to Yardly has a SR Q locomotor first produced in 1938, and the truck that slows Japp down before getting to the airport is a Bedford M-Type, a model first produced in 1939 (the specific one used in this episode is from 1947).
Continuity mistake: When Poirot enters the room of Mademoiselle Marvelle after her husband took off for the airport, she opens the door to him holding the door in two contrasting ways (right hand on the side of the door in the first shot, behind the door using the left hand to push it closed in the other). (00:41:35)
Factual error: The calendar behind Japp in Scotland Yard places November 13 as a Tuesday, but that is wrong for 1935, when the story is set (the whole season consistently goes with 1935 historical facts, Poirot in this episode makes a reference to "the Dutch", and the newspaper page Hastings shows up with at the end of the episode talks about the notorious gangster's death, happened that year). It was a Wednesday in 1935, to be Tuesday it should have been 1934. (00:34:00)
Factual error: At the end of the episode, during the fitting of the jacket with monsieur Fingler (last role of character actor Milo Sperber), Hastings is reading a newspaper prop sporting a doctored first page (the real headline, which was about Italy, is replaced by the fake PM speech) of The Daily Express based on the real Friday October 25th, 1935, which is wrong in month and day of the week (the appointment was on Thursday and it's supposed to be November going by the various calendars seen throughout the episode). (00:51:00)
Factual error: Poirot in this episode has to save the Prime Minister, kidnapped before he could attend the League of Nations Disarmament Conference. This episode is set in November (the previous of the season in October), but it can't possibly be 1935, since the Disarmament conference happened between 1932 and 1934. It makes mention of the PM being vital to have at least a chance at preventing Germany's rearmament, with Hitler withdrawing Germany from conference and League of Nations entirely in October 1933. It should be noted though that the conference happened in Geneva and not in Paris like it's described here. The original material did not have this sort of inconsistency since it was set to happen much earlier in the century and for the Treaty of Versailles, where Paris would have been the correct destination. Both short story and adaptation have as prime minister a fictional "David MacAdam", with him designed to be a stand-in for David Lloyd George, just recently replaced as PM.
Plot hole: The professional hitman from the US does not have a gun of his own, and does not bother in the long time elapsed between the apartment's hit and the nightclub (it could even be hours) to check if the gun he just grabbed is loaded or not. Who in the world would not take a split second of his time to peek at the cylinder and see how many bullets does he have? An assassin planning to kill with that gun, even.