Cotton was born in London on 20 June 1944, the son of Ester and Robert Norman Cotton. He trained at the Drama Centre, London.

For the Royal Shakespeare Company he has played leading roles in Granville Barker's The Marrying of Ann Leete, Henry VI, Edward IV, Richard III, The Plain Dealer, Some Americans Abroad by Richard Nelson, and as the Mayor in Brand by Ibsen in 2003.

In the West End he has starred in The Homecoming by Harold Pinter, Children of A Lesser God by Mark Medoff, Benefactors by Michael Frayn, An Ideal Husband by Oscar Wilde, Life x 3 by Yasmina Reza (RNT transfer), "Passion Play" (by Peter Nichols). He also played King Lear at the Southwark Playhouse and performed as Malvolio in a (historically correct all-male) production Twelfth Night at the reconstruction of Shakespeare's Globe. At the Old Vic, under the directorship of Kevin Spacey, he appeared as Seth Lord in The Philadelphia Story, as Northumberland in Richard II, and as Dr Finache in A Flea in Her Ear. In 2010 he played Henry IV in Shakespeare's Henry IV at the Globe Theatre. At Chichester he has appeared as Jim Casy in The Grapes Of Wrath in 2009 and as Arturo Santaniello in The Syndicate. In 2012, he played Victor Velasco in a major tour of Barefoot in the Park. In 2013 he played Jim in Passion Play by Peter Nichols at the Duke of York’s Theatre – and in 2014 starred as Billy in his own play Daytona at the Theatre Royal Haymarket.

Cotton worked extensively at the National Theatre Company during the period when Sir Laurence Olivier was its artistic Director. Cotton played leading roles in many productions including The Royal Hunt of the Sun, Rosencrantz and Guildenstern Are Dead, Much Ado in About Nothing, As You Like It, Peter Brook’s Oedipus, In His Own Write and many others.