Dwyer married his long term girlfriend Ann in 1972 at St. Patrick's Church in Grangetown. His best man at the ceremony was teammate Gary Bell. Dwyer and his wife moved into a flat on Cowbridge Road that was provided by Cardiff City, costing £5 per week. His wife worked as a secretary at the Angel Hotel in Cardiff. The couple's first child, Darren, was born in September 1974 and their second, Claire, was born in 1976. His wife, Ann, later became a care worker and spent 20 years in the role and also spent time working in a nursery.