The drama - penned by Who writer Mark Gatiss - will focus on the creation of the long-running BBC sci-fi drama in 1963.
Hartnell - the first actor to bring the Doctor to life - will be played by Harry Potter star Bradley.
"I'm absolutely thrilled," said the 70-year-old actor. "Mark has written such a wonderful script not only about the birth of a cultural phenomenon, but a moment in television's history.
"William Hartnell was one of the finest character actors of our time and as a fan I want to make sure that I do him justice. I'm so looking forward to getting started."
Bradley recently appeared in Doctor Who itself, playing space pirate Solomon in the episode 'Dinosaurs on a Spaceship', and is also known for appearing on Game of Thrones and in acclaimed BBC drama Our Friends in the North.
[Left: Jessica Raine, Right: Brian Cox]
Jessica Raine will also appear in An Adventure in Space and Time as Doctor Who's first producer Verity Lambert. The Call the Midwife star has already signed up for a role in the next series of Who airing on BBC One from March.
Rounding out the cast is Brian Cox (X-Men 2, Troy) as Sydney Newman, the Canadian producer who helped devise Doctor Who, and Sacha Dhawan (Last Tango In Halifax) as Waris Hussein, director of the show's first ever episode 'An Unearthly Child'.
"What a cast!" said writer and exec producer Gatiss. "I'm utterly delighted that everyone's favourite Time Lord will be in such brilliant and stellar company.
"We have a terrific team who can't wait to tell the fascinating and surprising story of how the Doctor began his journey through Space and Time."
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Watch a clip of William Hartnell in Doctor Who below: