Client Bug
This is a Bump from a previous post but i'm having the exact same problems.
Since the last patch, after the Champion Select screen, my game will not start. If I open up the window, it will show the typical Game in Progress screen, but the game itself will not start. If I close the window and relog, it shows the Click Here to Reconnect to Game button, but it takes a couple of tries in order for that button to actually reconnect to the game. This happens almost every time I try to play a game.