Come In, Your Time Is Up - S8-E4
Revealing mistake: In the very first scene Captain Mainwaring and Sergeant Wilson cast multiple shadows despite supposedly being in bright sunshine.
Revealing mistake: This actually happens throughout the entire show but is most obvious at the beginning of this episode: When they are in the churchyard it is obviously a studio stage, as there are shadows at all angles around the cast from the studio lights. Also, the lights are quite prominently reflected in Arthur Lowe's glasses.
Revealing mistake: When Frazer is attempting to get the chocolate out of the crane machine, he tells Mainwaring to keep quiet. When he does this, he takes his hands of the machine, and it picks up the chocolate on its own, without anyone controlling it.
Battle of the Giants! - S4-E14
Revealing mistake: Just before the two vans drive round in circles, when Mainwaring has trouble with the gearbox, the shot from inside the van shows a house through the driver's window. But in the exterior shots there are no buildings around, just trees.
Is There Honey Still for Tea? - S8-E3
Revealing mistake: When Mainwaring first comes into the bank, followed by the workmen with his new door, behind them is the 'high street', actually a painted backdrop. This is normal routine for studio sitcoms, however Mainwaring and the others all cast shadows on the backdrop, revealing it as a studio set.
Revealing mistake: When the knight moves back and forth during the clock mechanism's operation, as he swings his sword, a blue pullover sleeve is visible beneath his armour, revealing that he is an actor in costume and not an automaton.
Revealing mistake: The lights in the café are not working, despite the room being brightly-lit from above (i.e. the brightness cannot be attributed to sunlight entering through the window).
Answer: The show is a comedy, this was played for comedic effect and to show that in times of extreme fear, in this case about to be crushed by a steam train, the men had an adrenaline surge strong enough to pedal fast enough.