Continuity mistake: About 3/4 through the film, which is supposedly set in the 1900s, there's an exterior shot of Jenny Agutter (as Roberta) talking to William Mervyn (as "the old gentleman"). Over Mervyn's shoulder can be seen a white delivery van driving rather fast along a road on the other side of the valley from where the filming took place (this is very obvious in the wide-screen version but less noticeable in the pan-and-scan TV version).
Revealing mistake: When the 3 children are waiting for their mother, the rain is pouring down. Yet, when they are gathered around the Russian guy on the floor, their clothes are dry.
Continuity mistake: When the Grammar School boys are on the paper chase they run through a railway tunnel. The Railway Children go to the opposite end to see the grammar school boys come out, as they are waiting for the final boy to come through, the paper that is thrown on the ground for the boys to chase appears then disappears several times.
Other mistake: In the last scene where the children are walking away from the house after their father has gone in, in the background on the right hand side of the screen, a tree falls.
Other mistake: We see the old gentleman travelling this route several times during the course of the film. However he is always going the same direction! Does he teleport the return leg? We know this is a two way line as trains are seen going the other way, including the train father returns on.
Suggested correction: You say that the Railway Children only see "The Old Gentleman"/William Mervyn on a train going one way, but never on a return journey. Although, in both this film, and the original book, we are never wholly certain what he does (or did) for a living, or why he might be travelling on the train, it is implied that he is a person of some wealth and importance, who is respected by important people in London. Maybe he has important business to attend to in London, or elsewhere, that keeps him occupied until late in the day. Therefore he may return home by the last train, which may not come along the railway line until evening, or even at nighttime, by which time the Railway Children would be safe at home, possibly even asleep in bed.
Suggested correction: We don't see many things happen, that doesn't mean that they don't. His return journey just isn't relevant to the story.
Revealing mistake: In the scene where the children are stopping the train with the red petticoats, just before the train has completely halted, you can see the reverse effect used in the film. The train is actually reversing away from them for safety, and when reversed, the red petticoats are swaying with unnatural movements as they're being waved by the children.
Continuity mistake: When the children stop the train, the red petticoat Phyllis is waving quickly changes position between the shots when she is waving it, and when she stops waving it and runs towards the driver.
Continuity mistake: Although the name of the family in the film is established as "Waterbury" (in E. Nesbit's book the family doesn't have a surname), about 15 minutes into the film a man in the street says to Dinah Sheridan (playing the mother) "Good evening Mrs WaterFORD".

Deliberate mistake: When the family first entry Three Chimneys they have a single lamp as a light source. Entering from the door at the rear of the shot, everything in front of them should be in darkness to the viewer, but it's lit. Things in the background are lit even though there would be no line of sight to the light source. This is because lighting was added on set to allow the viewer to see things better, even though it ruined the realism.