I've seen some pretty convincing analysis that despite what both Clark and Bryan thought, it really wasn't Bryan who swung the convention, but Wilson's floor managers cutting deals and keeping Wilson from conceding -- which, according to William McAdoo (one of his leading campaign personnel and later Secretary of the Treasury and his son-in-law), he was on the verge of conceding the night before Bryan's thunderous speech. It still took over a day of ballots before Clark's lead was noticeably eroded.
Wilson's floor men had made several major deals at the convention: first, they got the resolution of the dispute over the rival Illinois delegations in their favor (there was an Illinois state boss, and a Chicago boss; the Chicago boss was allied with Hearst and, by extension, Clark); second, they struck deals with the Underwood camp in the interests of keeping the Southern Democrats from nominating Clark; and third, a deal was cut with Indiana, that shifted the ground to move the nomination out of Clark's grasp.
All of that, however, would have been for nothing had Wilson himself not stayed in at the convention. Once Clark had a majority of the votes, there was a call for Wilson to concede (Wilson having previously stated that the two-thirds rule was undemocratic). McCombs, Wilson's campaign manager, called Wilson and persuaded him to concede, releasing his delegates. When, several hours later, McAdoo found out what McCombs had done, he called up Wilson to persuade him otherwise -- that they still had a good chance and he should continue to fight it out. As it turns out, McAdoo was right.