Plummer's Relish appears to be a take on Branston Pickle, a popular condiment in Britain, dating back to the 1920's. Branston Pickle is used in sandwiches, as DCI Barnaby references, and is most commonly associated with the so-called "ploughman's lunch", a meal of bread, cheese, pickle and often ham. The rough similarity between "ploughman's" and "Plummer's" further reinforces the reference.
Anselm stresses the importance of Amelia signing the Power of Attorney in case she "goes doolally." Doolally is a colloquial phrase meaning mad or insane, and comes from the former British Army station Deolali, India. Soldiers were sent there to await ship home and often, due to heat, boredom and other stresses, exhibited signs of mental affliction known locally as the "Doolally Tap."
Bird-themed music is associated with Ralph Plummer. His ringtone is "Der Vogelfanger Bin Ich, Ja" from Mozart's "The Magic Flute." Another scene with Ralph in his duck blind opens with a theme which, if it is not "The Lark Ascending", by Vaughn-Williams, is a very good imitation.
Barnaby states one of the relish ingredients as Scotch Bonnet. If actually present it must have been in a small quantity because Scotch Bonnet is a pepper with a minimum Scoville rating of 100,000 (compared to the maximum of 8,000 for a typical Jalapeno). Any sizeable portion of the pepper in the relish would have made it extremely hot and spicy to consume.
James Fleet plays Ralph Plummer in this episode. He returns in episode 19.6, The Curse of the Ninth (2017), in the role of Michael Falconer.