"Mind you, I am the kind of bloke that shouts at the telly when a period drama uses the wrong kind of steam train - which is just about every time." I assume that Brian is a rail fan.

Using a 1958 Edsel in a movie set in 1955 would jump out at a car fan.

Being a musician, certain things jump out at me. In "Inside Llewyn Davis", Llewyn's character plays era appropriate guitars and music. They were even careful enough to show him with an old Hamilton capo early in the film, which made it bother me even more when he was later pictured using a Shubb capo.
A minor mistake was a Martin guitar with a black pick guard in the fifties in the Johnny Cash movie "Walk The Line".
Woody Guthrie was pictured playing a Mossman guitar in "Bound For Glory".
In the movie "Horse Feathers", Groucho uses a different guitar to play than the one he throws out of the canoe.
Horse Feathers Canoe Scene