At the age of five, Dickson contracted polio, but he was fortunate enough to make a complete recovery a year later. He attended Worksop College in Nottinghamshire playing Coriolanus in the Junior Play 1966. He graduated from the Guildhall School of Music and Drama and spent several seasons working in repertory theatres in Sheffield, Manchester, Leicester and Oxford among others.

In 1975, he made his West End debut in Pinero's The Gay Lord Quex, opposite Dame Judi Dench, which was directed by Sir John Gielgud at the Albery Theatre.

While playing Dean Rebel in Trafford Tanzi at London's Mermaid Theatre, he was spotted by the producers of the NBC mini-series A.D., who cast him in the lead role of Valerius. He spent the following nine months on location in Tunisia working opposite James Mason, Susan Sarandon, Ava Gardner and Ian McShane. Upon his return, he was cast in the eponymous role of James Bigglesworth in the British feature film, Biggles (American title: Biggles: Adventures in Time), which was selected as the Royal Premiere Film in 1986. He went on to star in several mini-series and the cult sci-fi series She-Wolf of London, which was given the alternate title Love & Curses when syndicated in the United States. In 2008, he starred in the film Chasing Chekhov, which won the first BAFTA LA Film Festival Award. He has recently completed work on Tim Burton's Alice in Wonderland.

TV credits include: I, Claudius, Secret Army, Blake's 7, Airline, Boon, Rockliffe's Babies, She-Wolf of London aka Love & Curses, Dynasty, Matlock, Baywatch, Sliders, Iron Man, Diagnosis: Murder, Alias, Mad Men and 1987 TV movie "The Murders in the Rue Morgue".

In 2011 Dickson played the eponymous role in Charles Dennis's award-winning film Atwill. He reprised the role of fugitive British Assassin Clive Atwill in the web series Atwill at Large and in the 2018 feature film Barking Mad.