He made his first team début on 18 October 2005 in the first round of the Football League Trophy against Halifax Town, which Bury lost 6–1, as a 16th minute substitute, replacing striker Tom Youngs, after keeper Craig Dootson had been sent off.
At the domestic level McGinlay has refereed 21 Old Firm derbies, during which he never sent off a player, but he considers his most nerve-racking match to be the Rangers–Aberdeen title decider match at the climax of the 1990–91 season.
In the following year, he was picked by manager Luis Milla for his squad that appeared in the 2012 Summer Olympics in London: in the first game against Japan, he was sent off late into the first half for bringing down Kensuke Nagai thus denying a clear goalscoring opportunity, in an eventual 0–1 loss.
Later he was selected as one of the English referees at the 1966 World Cup, gaining some notoriety there for his handling of the Uruguay versus West Germany quarter-final in which he sent off Horacio Troche and Héctor Silva.