Tarmey was born in Ardwick, Manchester, Lancashire. Shortly after his birth, he moved with his family to live in Bradford, Manchester, where he was also educated. Following the death of his father William in 1944 whilst driving an ambulance at the Battle of Arnhem during the Second World War, his mother Lilian remarried, to Robert Cleworth. Tarmey attended the Bradford Memorial School and the Queens Street School (which became the Philips Park Secondary Modern School). On leaving school, he was apprenticed to his stepfather, who was an asphalt spreader by trade. He also worked in the construction industry for a number of years.