Robson was born in South Shields, County Durham, of Scottish descent to a family of six siblings. Many of her forebears were engineers, mostly in shipping. Her father was a ship's engineer who moved from Wallsend near Newcastle to Palmers Green in 1907 and Southgate in 1910, both in north London, and later to Welwyn Garden City.
Her father discovered that Flora had a talent for recitation and, from the age of 5, she was taken around by horse and carriage to recite, and to compete in recitations. This established a pattern that remained with her.
Robson made her stage debut in 1921. By the 1930s she was appearing in several prominent films both in the UK and in Hollywood, alongside such stars as Laurence Olivier, Paul Muni and George Raft. Her most notable role was that of Queen Elizabeth I in both Fire Over England (1937) and The Sea Hawk (1940). In 1934, Robson played the Empress Elizabeth in Alexander Korda's Catherine the Great (1934). She was nominated for the Academy Award for Best Supporting Actress for her role as Angelique Buiton, a servant, in Saratoga Trunk (1945). The same year, audiences in the U.K. and the U.S. watched her hypnotic performance as Ftatateeta, the nursemaid and royal confidante and murderess-upon-command to Vivien Leigh's Queen Cleopatra in the screen adaptation of George Bernard Shaw's Caesar and Cleopatra (1945).
After the Second World 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 Sister Philippa in Black Narcissus (1947); as a magistrate in Good-Time Girl (1948); as a prospective Labour MP in Frieda (1947); and in the costume melodrama Saraband for Dead Lovers (1948). Her other film roles included the Empress Dowager Cixi in 55 Days at Peking (1963), Miss Milchrest in Murder at the Gallop (1963), the Queen of Hearts in Alice's Adventures in Wonderland (1972), and Livia in the aborted I, Claudius in 1937.