Thompson was born in Sleaford, Lincolnshire, the son of George Henry and Anne Thompson, and grew up Rudgwick, Sussex, attending Collyer's School, Horsham. He trained to be an actor at the Old Vic acting school in London and joined the Old Vic theatre company in 1952.

He worked mostly for the BBC, and was a presenter of the children's television programme Play School in the 1960s. He was best known as the narrator of The Magic Roundabout, for which he wrote the English-language scripts, using the visuals from the French Le Manège Enchanté. These were transmitted between October 1965 and January 1977. Thompson rarely worked on television after his voice became well-known, but occasionally appeared in programmes such as Doctor Who in the serial The Massacre of St Bartholomew's Eve in 1966.

Thompson was married to Scottish actress Phyllida Law whom he met while at the Old Vic, and was the father of actresses Emma Thompson and Sophie Thompson.

On 30 November 1982 Thompson died of a pulmonary embolism in London just three weeks after his 53rd birthday. His death was widely reported as he was well-known for his voice acting and narration.

In 1992, ten years after Thompson's death, fifty-two unseen episodes of The Magic Roundabout were broadcast on Channel 4. The adaptation and narration of the episodes was done by The Young Ones star Nigel Planer in a pastiche of Thompson's style.