Matthew Rhys' detective is shown smoking filter cigarettes which did not become commonplace until the late 1950s, but this is set in the 1930s.
In the 1930s, a British detective inspector who showed up for work in that unshaven state would be sent to the barber right away.
DS Miller mentions Audrey's parents were killed in a plane crash in Kenya. He pronounces it KEN-ya when in the 1930s it was pronounced KEENya (British colonial Kenya). The KEN-ya pronunciation was used after independence, well after WW2.
In the staircase scene, the correct form of the 1930s would have required a gentleman to wipe his mouth with a handkerchief afterwards before addressing a third party. Not doing so would have been considered slightly off hand. As Nevile Strange clearly had a good upbringing, and had a handkerchief on him at the time, he would never have made this faux pas.