In 2021, he was nominated for a Primetime Emmy Award for Outstanding Supporting Actor in a Comedy Series for his performance in Ted Lasso for his portrayal of Leslie Higgins.

[2] In the 2000s, Swift appeared in Gosford Park playing the footman Arthur, and Roman Polanski's Oliver Twist as Mr. Bumble.

In 2009, he played the lead in the true story of art forger Shaun Greenhalgh in The Antiques' Rogue Show[3] for BBC2 with Liz Smith and Peter Vaughn, The Deacon in a film adaptation of Anton Chekhov's short story The Duel and featured in Canoe Man, a 2010 TV drama based on the John Darwin disappearance case.

Swift is also a composer and his work includes the score for Werewolves: The Dark Survivors (Wide-eyed Entertainment) for the Discovery channel and ITV global.

In 2020, he played the role of Leslie Higgins in Apple TV+'s show, Ted Lasso, for which he received a Primetime Emmy Award nomination for Outstanding Supporting Actor in a Comedy Series in 2021.