She was born in South Shields, Durham, England and made her stage debut in 1921. Lacking the glamorous looks of a leading lady, she specialised in character roles, notably that of Queen Elizabeth I in both Fire over England (1937) and The Sea Hawk (1940). At the age of 32, Robson played the old Empress Elizabeth in Alexander Korda's Catherine the Great (1934).
After the war, demonstrating her range, she appeared in Holiday Camp (1947), the first of a series of films which featured the very ordinary Huggett family; as the Mother Superior in Michael Powell and Emeric Pressburger's Black Narcissus (1947); as a magistrate in Goodtime Girl (1948); as a prospective Labour MP in Basil Dearden's excellent Frieda (1947); and in Dearden's costume melodrama, Saraband for Dead Lovers (1948).
It was her success in Hollywood that brought her recognition, and in 1960 she was created a Dame of the British Empire. She died in Brighton.