Davidson was born in Riverside, California, the son of a Ghanaian father and English mother. He left the United States with his family for United Kingdom when he was two and a half years old, growing up in Hertfordshire, England.

Davidson is best known for his role as the transwoman Dil in the film The Crying Game (1992). For this role, he was nominated for an Academy Award for Best Supporting Actor, although the prize went to Gene Hackman for his performance in Unforgiven.

Director Neil Jordan revealed in the commentary to The Crying Game Collector's Edition DVD that Davidson was unaware, until the scene in which his character's transwoman identity was revealed was being filmed, that he would be fully nude in the film. Davidson also appeared as Ra in the film Stargate, (1994) his last film.

He no longer acts, preferring to work in the fashion industry. He has returned to the job he had before The Crying Game, working as a fashion assistant in London. He worked for Walt Disney Corporation and as fashion assistant for David and Elizabeth Emanuel.