At the beginning of the episode, Inspector Mallory is snoozing under a copy of The Big Sleep, by Raymond Chandler.
While many episodes of Father Brown reference other episodes from earlier seasons, "The Devil You Know" very presciently references two episodes which went on to feature in Season 7 of the show. When Inspector Mallory says, during the opening prologue: "... but for the part when asked to describe my involvement in the local community. Preferably something that didn't involve bell-ringing, rambling, children or any other form of undue exertion", he is referencing S7 E2 "The Passing Bell" (about a possible murder in the St Mary's bell-ringing cloister) and S7 E9 "The Skylark Scandal" (in which the whole crew go on a long weekend ramble to the Skylark hostel in the countryside).
"The Devil You Know", like all episodes of Father Brown, is set in the year 1953. Therefore, when Shirley Krieger tells Roger Frobisher at the end of the show that "there is a new [play by] Arthur Miller on in London", she is referring to The Crucible, which had its world premiere on Broadway on January 22 1953. However, the UK premiere season of The Crucible did not take place until 1954, and even then was held at the Bristol Old Vic rather than in London.