Dudgeon was born and brought up in Doncaster, West Riding of Yorkshire and attended Intake Secondary Modern school, among others. He acted in several school plays (including Rosencrantz and Guildernstern are Dead) and went on to study drama at the University of Bristol (1979–82). He made his first screen appearance in 1987. The following year he appeared as a Second World War pilot in Piece of Cake, alongside Tim Woodward, Jeremy Northam and Nathaniel Parker.
Dudgeon is married to BBC Radio producer Mary Peate. They have two children.