Born in Doncaster, West Riding of Yorkshire, Concannon was educated at Rossington Secondary School and through the Extra-Mural Department of the University of Nottingham. He worked as a miner and as a National Union of Mineworkers (NUM) official. He was a councillor on Mansfield Borough Council from 1963.
Concannon was elected as the Member of Parliament for Mansfield at the 1966 election. Under Harold Wilson and James Callaghan (Prime Minister from 1976 to 1979), he served as a government whip and as Northern Ireland minister, and was appointed a member of the Privy Council in 1978. He was sponsored by the NUM.
Concannon died in Mansfield on 14 December 2003, aged 73.