The King Was in His Counting House - S5-E7
Factual error: Captain Mainwaring is transferring money from his bank to another branch after it has been bombed. He has the money in a basket and is transporting it on a horse and cart when some of the money starts blowing out of the basket. Pike, Wilson, Walker, and Jones are riding bicycles behind him. They stop to pick up the money and just before Jones tells Pike to ride after Mainwaring to tell him about the money, you can see a silver car parked on the left hand grass verge in the direction Mainwaring is heading. This car was obviously made long after World War Two finished.
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.