Dwyer attended Bishop Mostyn High School in Ely as a teenager, but later admitted to having little interest in pursuing his academic studies, choosing to focus on his sporting activities. He played rugby during midweek and football on Saturdays, sometimes playing two matches in one day, in the morning for his school's team and in the afternoon for North Clive Street Youth Club. In summer months, he also played baseball for Cardiff Catholics. Dwyer met his future wife Ann in his teens at a local nightclub and the pair began dating soon after.