Cox was born on 1 June 1946 in Dundee, Scotland. He studied at the Dundee Repertory Theatre and at the London Academy of Music and Dramatic Art. Cox was married to Caroline Burt from 1968 until they divorced in 1986. He married Nicole Ansari in 2002. With Burt, he had four children, two who were stillborn. With Ansari, he has two sons.