Dwyer became affectionally known as "Joe" by Cardiff fans and teammates, a nickname that originated during his early years at the club when goalkeeper Fred Davies commented that Dwyer resembled Everton forward Joe Royle. As an apprentice, he was also employed as the club's ground staff where youth players were required to fulfill various tasks, such as cleaning kit and clearing rubbish from the stands at the ground after matches. With no organised age group league to play in, the youth team competed in local amateur divisions against senior teams. During the 1970–71 season, Cardiff's youth team entered the FA Youth Cup and reached the final of the competition, the furthest the side has ever advanced. Having won five ties, Cardiff faced Arsenal in the final. Cardiff held Arsenal to a goalless draw in the first leg before losing the return fixture 2–0 at Ninian Park.