Runners can leave base as soon as the ball is touched by any defensive player. Not just touched by the glove and not just if the ball is caught by the same player that bobbled it.
Suppose the batter hit a line drive up the middle. If the ball barely nicked the pitcher (glove, body or uniform), then continued on the centerfield, where it was caught by F8, the runner could leave the base when the ball nicked the pitcher. (Batter would still be out on the catch.)
This question comes up a lot every year. The rule was first put into place in the very early days of baseball. Prior to this rule, an outfielder could purposely juggle a fly ball, without actually catching it, while running all the way toward the infield. Back when the runners had to tag up until the ball was actually caught, this tactic would prevent the runner from advancing on a fly ball. So, the rule was changed to when the ball was first touched, instead of when it was caught.
