Nagy moved to London in 1992, where her playwriting career began in earnest at the Royal Court Theatre under the artistic direction of Stephen Daldry for whom she served as the Royal Court's writer-in-residence in the mid-1990s.

Nagy's plays have been performed all over the world and include Weldon Rising, first produced by the Royal Court Theatre in association with the Liverpool Playhouse in 1992; Butterfly Kiss, first produced by the Almeida Theatre Company in 1994; The Scarlet Letter, an adaptation of Nathaniel Hawthorne’s classic novel, commissioned and first produced by the Denver Centre Theatre in 1994; Trip's Cinch, commissioned and first produced by the Actors Theatre of Louisville in 1994 and received its UK premiere in 2002; The Strip, commissioned and first produced by the Royal Court Theatre in 1995; and Disappeared, a joint winner of both the 1992 Mobil International Playwriting Prize and the 1995 Susan Smith Blackburn Prize. Disappeared was premiered at the Royal Court in 1995 in a production directed by the author which subsequently toured the UK before a London run at the Royal Court Theatre. The play went on to win the Writers Guild of Great Britain Best Regional Play award and the Eileen Anderson/Central Television Best Play award. In February 1999 Disappeared was presented at the Steppenwolf Theatre, Chicago by RoadWorks Productions.

Nagy's most recent plays are Never Land, which premiered at the Royal Court Theatre in January 1998 in a co-production with The Foundry; and The Talented Mr Ripley, adapted from the novel by Patricia Highsmith which premiered at the Palace Theatre Watford in October 1998 and later produced by the Melbourne Theatre Company in February 1999. Her version of Chekov’s The Seagull was produced at Chichester Festival Theatre in the summer of 2003. In 2005 she directed a production of The Scarlet Letter at the same venue.

She is developing two films for Film Four: an adaptation of Patricia Highsmith's The Price of Salt and an adaptation of Kate Atkinson's Case Histories.

She was nominated for Emmy Awards for writing and directing Mrs. Harris (2005), her screen debut. The film starred Ben Kingsley and Annette Bening (both also Emmy nominated), and garnered a total of 12 Emmy Award nominations, 3 Golden Globe nominations, 3 Screen Actors Guild Award nominations and numerous other media awards. It premiered on HBO in February 2006.