Wilson, 20, and a team including Louis Smith and Max Whitlock finished less than two points down on China, who claimed bronze behind new Olympic champions Japan and Russia in silver medal position.
Smith slipped off the pommel horse as Great Britain missed out on a medal.
The British team, who had held third place for most of the competition, faced a huge task on the final rotation after being nudged into fourth by reigning champions China.
And the pressure told on Smith, on his only apparatus of the evening, as he came off midway through his routine to score 14.766 and effectively end their slim chances of nudging back in on the medals.
Even a strong finale from world champion Whitlock, scoring 15.991, could not rectify the situation as the Britons - also including Brinn Bevan, Kristian Thomas and Yorkshireman Wilson - were forced to settle for fourth place.
China had hardly figured until a stunning parallel bars performance in the penultimate routine was enough for bronze.
Wilson, 20, from Pudsey in Leeds, excelled on the high bar for Team GB.
Later this week, Wilson, the two-time Commonwealth Games gold medallist contests the individual high bar and all-round finals.