Slattery was born in Stonebridge, London, to Irish Catholic immigrant parents. He attended Gunnersbury School in Brentford. In his youth, he represented England at under-15 judo. He studied Mediaeval and Modern Languages at Trinity Hall, Cambridge University, where he also became a member of the Cambridge Footlights. Like his contemporaries Stephen Fry and Hugh Laurie, he became a staple of the British television comedy circuit.

Slattery first broke into television as a regular performer on Chris Tarrant's follow up to "O.T.T.", "Saturday Stayback". He was a regular on Whose Line Is It Anyway?, starred in his own improvisational comedy series S&M alongside Mike McShane, and has appeared on other panel quizzes such as Have I Got News For You. He was a regular on the TV version of the quiz show Just a Minute and was also on the radio version several times, including the Live version held at the Edinburgh Festival.

As a serious actor he has appeared in The Crying Game, To Die For, Peter's Friends, and The Wedding Tackle. He appeared on the London stage in the musicals Me and My Girl and Radio Times and in the play Neville's Island.

At the end of the 1980s, he became a film critic, presenting his own show on British television, Saturday Night at the Movies. He also appeared in the ITV sitcom That's Love with Jimmy Mulville. Other TV appearances include The Music Game alongside good friend Richard Vranch, Just a Gigolo, and as a regular guest with both Ruby Wax and Clive Anderson.

He has also been a regular guest with The Comedy Store Players, both at The Comedy Store in London and out on tour around the country.

Early in the 1990s he became over-exposed as a celebrity, so much so that he became the target of satire. For example, the Have I Got News For You 1991 annual showed images of the game from around the world, and each local variant had Tony as a guest. Spitting Image showed a sketch where an anthropomorphised BBC2 logo refused to have blue paint splattered on it and Tony stepped in for publicity's sake. The satirical magazine Private Eye once published a memorable cartoon depicting Tony's answering machine with the outgoing message "Yes, I'll do it!".

In 1992 he appeared in the film Carry On Columbus.

In the mid-1990s he suffered what he described as a 'mid-life crisis' which culminated in 1996 with a six-month period of reclusiveness, during which he did not answer his door or telephone, "or open bills, or wash," he says; "I just sat." Eventually, one of his friends broke down the door of his flat and convinced him to go to hospital.

Now recovered, he has returned to presenting on British cable television. In January 2005 he appeared in the TV movie Ahead of the Class with Julie Walters. In December 2005 he joined the long-running drama Coronation Street as Eric the Bookie and in April 2006 is a contributor to Grumpy Old Men on BBC Two.