Shaughnessy was born in London, the son of Alfred Shaughnessy, a television writer who was the scriptwriter for Upstairs, Downstairs, and the actress Jean Lodge. His brother is another actor and scriptwriter, David Shaughnessy. He is of Irish and English descent. Having been born into a show-business family, he started appearing in plays during grade school. After attending Eton, he read for a law degree at Cambridge University.

While at Cambridge, he joined the famous Footlights club. After graduating, he decided to return to acting and enrolled in a London drama school, which led to him touring in a repertory company. He moved to America to follow the actress whom he would eventually marry, Susan Fallender.

Shaughnessy is known for playing Shane Donovan on Days of our Lives from 1984 to 1992. His character's romance with Patsy Pease's character, Kimberly Brady, made the duo into a soap supercouple and revived interest in Days among teenage viewers in the 1980s.

Shaughnessy is also well known for his portrayal of Maxwell Sheffield opposite Fran Drescher on The Nanny from 1993 to 1999. He and Drescher resumed acting together on Drescher's latest sitcom, Living with Fran, wherein Shaughnessy appeared fairly frequently as her philandering but needy ex-husband, Ted.

Living with Fran was canceled on May 17, 2006, after two seasons. Neither Living with Fran nor The Nanny have ever been shown on British terrestrial television and as a result, despite his "very British" persona, Shaughnessy is virtually unknown in the UK itself. The Nanny airs at 7 am on UK Living TV on the weekends. So far, all 6 seasons have been shown.

Shaughnessy also appeared in the Disney Channel 2002 made-for-TV movie Get a Clue. He was also in the Halloween 2000 made-for-TV movie Mom's Got A Date With A Vampire starring opposite Caroline Rhea. He played Alec Colson in the 8th-season episode "Covenant" of Stargate SG-1. He also voiced Dr Quintaine in the PC game Freelancer.

He has recently appeared on an episode of Law & Order: Special Victims Unit and is also the voice of Dennis the Goldfish on the Disney Channel cartoon series Stanley.

On May 11, 2002, Shaughnessy won an Emmy Award for Outstanding Performer in an Animated Program for his portrayal of Dennis the Goldfish on Stanley. He has also appeared several times on Sabrina the Teenage Witch.