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.
Factual error: The motorbike and sidecar used by the verger is a post-war 1945 BSA M21 with sidecar. The period is supposed to be 1942.
Continuity mistake: In this episode, specifically during the scene set in the Tea Room, Mr Cheeseman is holding a newspaper from The Daily Sketch - dated June 19th 1940. However, during an earlier episode (S3-E6 Room at the Bottom), George Mainwaring was spared from showing his demotion by the sinking of the Bismarck, which occurred on 26th May 1941. The newspaper held by Cheeseman in the Tea Room was at least a year old by that point.
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.