Although born in London, Cowan was brought up in Colchester, Essex. He is the son of a consultant physician and a charity worker, and has a younger brother and sister. Cowan boarded at Uppingham School in Rutland. He later obtained a first class degree in Drama at the University of Birmingham, before attending the Royal Academy of Dramatic Art in London, from which he graduated in July 2001. From 1994 to 1996, Cowan was a member of the National Youth Music Theatre. He plays guitar and cello, and has worked with the London Sinfonia. He lives in Dalston, London. On 7 August 2011, Elliot swam the 16 miles distance of Lake Zurich to raise funds for Guy's and St. Thomas's Neonatal Unit, in memory of Alfie Blacksell.