Shaw was born in County Cork and was raised in the Roman Catholic faith. Her father, Denis Wilson, was an ophthalmic surgeon and her mother, Mary, was a physicist.
She attended secondary school at Scoil Mhuire in Cork City. She received her degree in University College Cork. She trained at the Royal Academy of Dramatic Art (RADA) in London and was part of a 'new wave' of actors to emerge from the Academy. She received much acclaim as Julia in the National Theatre production of Richard Sheridan's The Rivals (1983).
Shaw's theatrical roles include Celia in As You Like It (1984), Madame de Volanges in Les Liaisons Dangereuses (1985), Katherine in The Taming of the Shrew (1987), Lady Franjul in The New Inn (1987), Young Woman in Machinal (1993), for which she won the Laurence Olivier Award for Best Actress, Winnie in Happy Days (2007), and the title roles in Electra (1988), The Good Person of Sechuan (1989), Hedda Gabler (1991), The Prime of Miss Jean Brodie (1998) and Medea (2000). She performed T. S. Eliot's poem The Waste Land as a one-person show at the Liberty Theatre in New York to great acclaim in 1996, winning the Drama Desk Award for Outstanding One-Person Show for her performance.
She played Miss Morrison in the 1984 The Adventures of Sherlock Holmes episode "The Adventure of the Crooked Man" and Catherine Greenshaw in Agatha Christie's Marple episode "Greenshaw's Folly" in 2013.
Shaw notably played the male lead in Richard II, directed by Deborah Warner in 1995. Shaw has collaborated with Warner on a number of occasions, on both stage and screen. Shaw has also worked in film and television, including My Left Foot (1989), Mountains of the Moon (1990), Three Men and a Little Lady (1990), Super Mario Bros. (1993), Undercover Blues (1993), Persuasion (1995), Jane Eyre (1996), The Butcher Boy (1997), The Avengers (1998), Gormenghast (2000), and five of the Harry Potter films in which she played Harry Potter's aunt Petunia Dursley. Shaw had a brief but key role in Brian DePalma's The Black Dahlia (2006).
In 2009, Shaw collaborated with Deborah Warner again, taking the lead role in Tony Kushner's translation of Bertolt Brecht's Mother Courage and Her Children. In a 2002 article for The Daily Telegraph, Rupert Christiansen described their professional relationship as "surely one of the most richly creative partnerships in theatrical history." Other collaborations between the two women include productions of Brecht's The Good Woman of Szechuan and Ibsen's Hedda Gabler, the latter was adapted for television.
Shaw appeared in The Waste Land at Wilton's Music Hall in January 2010 and in a National Theatre revival of London Assurance in March 2010. In November 2010, Shaw starred in Ibsen's John Gabriel Borkman at the Abbey Theatre, Dublin alongside Alan Rickman and Lindsay Duncan. The play was also staged in New York's Brooklyn Academy of Music in 2011.
Shaw appeared in season four of American TV Show True Blood. Shaw's character, Marnie Stonebrook, has been described as an underachieving palm reader who is spiritually possessed by an actual witch. Her character leads a coven of necromancer witches who threaten the status quo in Bon Temps, erasing most of Eric Northman's memories and leaving him almost helpless when he tries to kill her and break up their coven.
In 2012, Shaw appeared in the National Theatre revival of Scenes from an Execution by Howard Barker.
The world’s largest solo theatre festival, United Solo recognized her performance in The Testament of Mary on Broadway with the 2013 United Solo Special Award.
Shaw will portray Countess Constance Markievicz, an Irish nationalist and first woman to be elected to the British Parliament, in the centennial commemoration biopic film The Rising, written and produced by Kevin McCann. The film was set to be released on St Patrick's Day, 2016.When Love Speaks (2002, EMI Classics): "It is thy will thy image should keep open"
Simon Schama's John Donne: 2009