Grout spent many decades on the London stage, appearing in such shows as Twelfth Night, The Mousetrap, The Lily White Boys, Sweet Bird of Youth, Man and Superman, When We Are Married, The Way of the World, Quartermaine's Terms, and Mary Stuart.

He was best known for his television work, including his roles in All Creatures Great and Small, Rumpole of the Bailey, Yes Minister, Inspector Morse, and David Copperfield. In addition, he played Charley in a British telefilm of Arthur Miller's Death of a Salesman and the title role in a television production of Galileo.

Grout also co-starred in two BBC radio series, King Street and Old Harry's Game, and appeared in numerous films, including The Ruling Class.