Born in Amersham, Buckinghamshire, Godley began his acting career at the age of 9, in a BBC radio production of Hemingway's My Old Man. His first stage role came at age 11, as Prince Giovanni in The White Devil at The Old Vic. His childhood career also included work at the National Theatre, in Lillian Helman's Watch on the Rhine, and Close of Play, directed by Harold Pinter. Godley achieved national prominence after playing the lead in the 1984 BBC TV adaptation of J. Meade Falkner's Moonfleet.

In 1986 Godley joined Alan Ayckbourn's theatre company in Scarborough, where he stayed for three seasons. Productions included June Moon and The Revengers' Comedies, both of which transferred to the West End, and Mr A's Amazing Maze Plays, which transferred to the National Theatre.

He spent one season as a member of the Royal Shakespeare Company, before creating the role of Cliff in Sam Mendes's production of Cabaret at the Donmar Warehouse. Several major productions followed, including Mouth to Mouth at the Royal Court, The Front Page at the Donmar, and The Rivals in the West End. after which Godley went on to create a series of roles at the National, including Kenneth Williams in Cleo, Camping, Emmanuelle and Dick; Michal, in Martin McDonagh's The Pillowman, and the title role in Howard Davies' production of Paul. He also played Jack Worthing in the West End production of The Importance of Being Earnest and Victor in both the West End and 2001 Broadway production of Noël Coward's Private Lives. He played Lord Evelyn Oakleigh in Roundabout Theatre's multi award-winning Broadway revival of Anything Goes, and was Raymond Babitt in the West End production of Rain Man opposite Josh Hartnett. He was most recently seen in the lead role in a new adaptation by Dennis Kelly of Georg Kaiser's 1912 German expressionist masterpiece, From Morning to Midnight, at the National Theatre.