Cochrane was born in Brighton, East Sussex. He was educated at Cranleigh School. He has had many television and radio roles including Oliver Sterling in the Radio 4 soap opera The Archers, The Pallisers (1974), Wings (1977–78), Love in a Cold Climate (1980), The Citadel (1983), a BBC serial adaptation of Goodbye Mr. Chips (1984), No Job for a Lady, The Chief (1990–1995), and as Sir Henry Simmerson in the Sharpe series. His film career has included roles in Escape to Victory (1981), The Return of the Soldier (1982), Real Life (1984), Number One Gun (1990), The Saint (1997), Incognito (1998), A Different Loyalty (2004) and The Iron Lady (2011).