Morton began her career as a child actor. At 13 she joined the Central Junior Television Workshop. At 16 she moved to London and appeared in plays at the Royal Court Theatre, also securing an early break as a guest star in an episode of Cracker. Although she applied to institutions such as RADA, she received no further formal training. She went on to star in television productions like Jane Eyre and The History of Tom Jones, a Foundling.

Starring roles in Band of Gold (1995), and the film Under the Skin in 1997, raised her profile in the UK. The latter impressed Woody Allen so much he cast her in his film Sweet and Lowdown for which she would receive a nomination for the Academy Award for Best Supporting Actress in 1999. She is the first actress nominated for the Oscar for roles in a film in which she does not utter even a single word. In 2002 she won the British Independent Film Award for Best Actress for her work in Morvern Callar, followed by a nomination for the Academy Aw