Slattery was born in Stonebridge, London, to Irish Catholic immigrant parents. In his youth, he represented England in under-15 judo. He was educated at Gunnersbury Boys Grammar School in West London, and later studied Modern and Medieval Languages, specialising in French and Spanish, at Trinity Hall, Cambridge, where he became the president 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 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 on tour.

Early in the 1990s he became over-exposed as a celebrity, to the extent that he was a 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 featured Slattery as a guest.

Spitting Image showed a sketch where an anthropomorphised BBC2 logo refused to have blue paint splattered on it and Slattery intervened for the sake of publicity. The satirical magazine Private Eye once published a memorable cartoon depicting his answering machine with the outgoing message "Yes, I'll do it!"

In 1992 he appeared in the film Carry On Columbus. In the same year, he appeared in the series Dead Ringer, filmed for the observation round in The Krypton Factor. It was during this period that Slattery also appeared in the BBC sci-fi comedy series Red Dwarf in the episode "Kryten".

Slattery here played the voice of the main character on Kryten's favourite soap opera, "Androids" (a parody of Neighbours). He also reappeared in Red Dwarf in 1999 as the voice of the vending machine which threatens Arnold Rimmer in the final episode of the series, "Only the Good...".

Also in 1992, he appeared as a contestant on Channel 4's now defunct show GamesMaster, in which he said on the show he hated video games, despite the show being entirely devoted to them. He played the real-time arcade shooter Who shot Johnny Rock?, failing the challenge by shooting an innocent victim in the game.

In 1993 he starred in the ITV sitcom Just a Gigolo. Only one series was ever made.