When James and Helen are eating ice cream in Whitby, ice cream is clearly visible on James's top lip. After he has turned round, all traces of ice cream have vanished.
Early on (and whilst James is working at the surgery), Siegfried is seen reading the "Daily Express" newspaper dated Saturday June 5th 1937, but James tells Helen in a later scene that he's only been at the practice since July.
When the group wedding photograph is being taken, the bride's father, Mr Alderson, is not there yet he is seen when the couple drive away. The bride's father would definitely be posing with the others for the photograph.
Tricki Woo invites Mr Herriot to a garden party to be held on 5 August, but the film is set in 1937 which would make that a Thursday, a most unlikely day for a garden party.
During the "Lambeth Walk" dancehall sequence, several men can be seen sitting at tables in the background wearing jeans, t-shirts and with long hair. The film is set in 1937.