Simpson was born in Leamington, Warwickshire. His father, Robert Warren Simpson, was a descendant of Sir James Young Simpson, the Scottish pioneer of anaesthetics; his mother, Helena Hendrika Govaars, was the daughter of Gerrit Govaars, founder of the , the Dutch arm of the Salvation Army. Simpson studied at Westminster School. He was intended for a medical career and studied in London for two years before his determination to be a musician gained the upper hand. A conscientious objector in World War II, he served with an ARP mobile surgical unit during the London Blitz, while taking lessons from Herbert Howells. Howells persuaded him to take the Durham University Bachelor of Music degree, and in 1952 he gained the further degree of Doctor of Music from that university, the submitted work being his First Symphony. After the war Simpson lectured extensively and founded the Exploratory Concerts Society; in 1951 he joined the music staff of the BBC and became one of its best-known and most respected music producers, remaining with the corporation for nearly three decades. Simpson was a great champion of Havergal Brian's music, and under the BBC's auspices he produced many broadcasts featuring Brian's works. These included the mammoth "Gothic" Symphony in 1966 under Sir Adrian Boult, and in 1973 the 28th Symphony under Leopold Stokowski who, at the age of 91, was premiering a work written by a 91-year-old composer.