Canny began his career as a theatre director and writer. He was associate director for theatre company Complicite from 1999–2004 where productions included Mnemonic, The Noise of Time, The Elephant Vanishes, Light and he was also Associate Director on Al Pacino's The Resistible Rise of Arturo Ui. He was the dramaturg on Complicite's production of Measure for Measure at the National Theatre.
In 2009 he won the Sony Gold Drama Award with Mr Larkin's Awkward Day, a comedy radio play by Chris Harrald, broadcast on BBC Radio 4 on Tuesday, 29 April 2008 as the Afternoon Play. The Sony Gold citation said: "Assured direction, excellent performances and concise, skilfully-researched writing all made this deceptively straightforward story a masterpiece. Funny and touching by turns, a single, seemingly insignificant incident in the life of Philip Larkin brought out the humanity and humour of a poet whose personal life is not commonly associated with either."
In 2010 he won the Sony Gold Drama Award with People Snogging in Public Places by Jack Thorne. It was broadcast on Radio 3's The Wire. The Sony Gold citation said: "The judges described this as a wonderfully written and performed, highly original piece of radio drama in which the production perfectly mirrored the subject. Painful and funny, it was a bold exciting listen." The production was also shortlisted for the 2010 Prix Italia and the Tinniswood Award.
Canny was listed as the Executive Producer of the winners of the Gold, Silver and Bronze Sony Awards for Best Comedy in 2011. He was Executive Producer of the pan-BBC initiative for the Edinburgh Festival in 2011.
Canny was Executive Producer of more than 300 hours of comedy on Radio 4 including: News Quiz, Tom Basden's Party, Rudy's Rare Records, Radio 4's Advent Calendar and The Show What You Wrote.
Plays written include: A Dulditch Angel, Spyski (revived as The Importance of Being Honest), Origins, No Wise Men, The Hound of the Baskervilles and The Arthur Conan Doyle Appreciation Society.
By Spring 2019 The Hound of the Baskervilles has had more than 130 theatrical productions worldwide with the majority being in the US.
In September 2013, the BBC announced that they had appointed Canny as Executive Producer, BBC Comedy Scotland. There he has been the Executive Producer of Miller's Mountain (winner Scottish BAFTA Best Comedy 2014), Bob Servant (winner Best Comedy RTS Scotland Award), Don't Drop the Baton, Comedy Playhouse: Where it All Began, What's Funny about the Indyref, Still Game Live, Radges, Sketchland, Two Doors Down (winner Best Comedy RTS Scotland Award), Mrs Brown's Boys, Gary: Tank Commander Election Special, Stop/Start, Porridge, Pumped, British Sitcom: 60 Years of Laughing at Ourselves, A Christmas Carol Goes Wrong, Lost Sitcoms and Still Game series 7, 8 & 9.
- Staff (12 May 2009). "Sony Radio Academy Awards 2009: the winners – Telegraph". The Daily Telegraph. London: TMG. ISSN 0307-1235. OCLC 49632006. Retrieved 3 May 2011.
- Staff (2011). "The Sony Radio Academy Awards – Winners – 2010 – Production Awards – Best Drama". radioawards.org. Retrieved 3 May 2011.
- BBC Edinburgh Fringe – 2011 – Home
- BBC – Suffolk – Entertainment – A Dulditch Angel
- PEEPOLYKUS: SPYSKI! (The Importance of Being Honest)
- PEEPOLYKUS: No Wise Men
- Sommers, Michael (12 May 2012). "A Review of 'The Hound of the Baskervilles,' at the New Jersey Repertory Company". The New York Times.
- BBC comedy lures back Princess exec | News | Broadcast
- Mountain Goats – BBC1 Sitcom – British Comedy Guide
- British Academy Scotland Awards: Winners in 2014 | BAFTA
- Bob Servant
- Don't Drop The Baton – BBC1 Scot Sketch Show – British Comedy Guide
- Comedy Playhouse: Where It All Began (TV Movie 2014) – IMDb
- BBC Two – What's Funny About the Indyref?
- Still Game – Live in Glasgow [DVD]: Amazon.co.uk: Ford Kiernan, Greg Hemphill, Michael Hines, Pati Marr, Laura Marks: DVD & Blu-ray
- Radges – BBC3 Sitcom – British Comedy Guide
- BBC Two – Two Doors Down
- Nicola Sturgeon gets election grilling – from Gary: Tank Commander – The Scotsman
- Stop/Start – BBC1 Sitcom – British Comedy Guide