Elliott was born in London and made his film debut in 1949. He went on to play a large range of parts, generally specialising in the "typical" English gentleman. He appeared in films as diverse as Alfie, Raiders of the Lost Ark, Brimstone and Treacle and A Room with a View (1986) - he was nominated for an Academy Award for his supporting role in the latter.
In 1988, Elliott was awarded the CBE for his services to acting.
Although gay, Elliott was married twice (the first time to the British actress, Virginia McKenna) and had children. He died of AIDS.