This bug has happened twice to me before. It's been around for a really long time, but there isn't a way to recreate it. The first time, people began to leave, but the game did not disband, suggesting that the league client had already launched the game. The second time, after an extensive wait, the game eventually launched.
Honestly, I would be upset with Riot about the length of time this bug has been around, but considering the fact of how rare it is, and how little it can happen. It reminds me of the current Tryndamere bug, where he is occasionally able to kill allies with his spin. Thankfully, only a handful of players can vouch for this bug as well.
On the bright side, this weird bug is more comical than annoying in most cases. Everyone sits around expectantly but they never get in the game.
My best answer is to leave and log back in. Sometimes it will let you reconnect to the match, and people just join in slowly. It isn't exactly fair, but it's the best you can do.
