Coote was born in London and educated at Hurstpierpoint College in Sussex. He began his stage career at the age of 16, performing in Britain, South Africa, and Australia before arriving in Hollywood in the late 1930s. He played a succession of pompous British types in supporting roles, including a brief but memorable turn as Sgt. Bertie Higginbotham in Gunga Din (1939). His acting career was interrupted by his Service as a squadron leader in the Royal Canadian Air Force during World War II. He played Bob Trubshawe in Powell and Pressburger's A Matter of Life and Death (1946), chosen for the first-ever Royal Film Performance on 1 November 1946, before he returned to Hollywood, where his films included The Ghost and Mrs. Muir (1947), Forever Amber (1947), The Three Musketeers (1948), and Orson Welles' Othello (1952).