I am writing a book set in England in 1938 - 39. I am wondering what radio programs would the average person be listening to in London England in 1938 - 39.

Brits were listening to the BBC news events of WWII - Hitler's marches and consquests, listening to his speeches and terror. They were also listening to Winston Churchill.
They alos listened to sports - Cricket and the commentaries. Orson Welles' production of, "War of the Worlds" was heard over in 1938 or 1939. England, over the Red Network of the National Broadcasting Company, listened to Charlie McCarthy.