Davison was born Peter Moffett in London, his father being originally from Guyana. He studied at the Central School of Speech and Drama, and appeared in several stage productions and some minor television roles before he got his big break in 1978. His performance as the ne'er-do-well Tristan Farnon in All Creatures Great and Small made him a household name. He married American actress Sandra Dickinson in the same year, but they divorced in 1994. He and Dickinson appeared together in an episode of '\'The Hitch-Hiker's Guide to the Galaxy''.
Davison appeared in some British sitcoms before signing a contract to play the Doctor for three years, succeeding Tom Baker. Attracting such a high-profile actor was as much of a coup for the programme's producers as getting the role was for him, but the contract was not renewed because he feared being typecast.
It was not until 1986 that Davison worked on another really popular series. He played Dr Stephen Daker, the ingenuous hero of A Very Peculiar Practice, written by Andrew Davies. The surreal comedy-drama was revived several years later as A Very Polish Practice. Davison also played the lead in Campion, a series based on the period whodunnits of Margery Allingham. This, and the opportunity to play Tristan Farnon again in 1985 and 1990, kept Davison busy until the early 1990s, when he gradually faded from the public eye. He continued to appear occasionally on television, but it was not until 2000 that he returned in another major role, that of David Braithwaite in At Home with the Braithwaites. In 2003, he gained the title role in the series The Last Detective.