Alfred Thompson was born in Bangalore, India, to British civil servant George and his Irish wife, Florence Green. Deaf from birth, he attended the Royal School for Deaf Children in Margate, where he learned sign language. At 13, he was transferred by his father to a private oral school in Brondesbury due to concerns about his speech. Later in life, the press referred to him as the "deaf and dumb" artist.

Thomson briefly attended the London Art School in Kensington but, after failing the Royal Academy School entry exam, his father sent him to work on a farm in Kent, forbidding him from pursuing art. However, Alfred later left the farm and found his first paid work designing posters for a whisky company at Vitagraph in Long Acre and for Daimler Cars. At the end of the First World War Thomson established himself as a commercial artist.

During Second World War II Thomson completed several commissions for the War Artists' Advisory Committee and, in September 1942, became a full-time salaried artist attached to the Air Ministry, succeeding Eric Kennington in the role. He painted numerous portraits of RAF aircrews, as well as medical and civil defence subjects.

He also became the last  person to win an Olympic Gold Medal for painting (in 1948 London Olympic Games), as medals for art were abandoned in subsequent Olympic games.