[Client] Game doesn't start (stuck in champion selection)

On the last week or two I've encountered the following bug a few times:

The client stays in champion selection after the countdown finish and the game doesn't start (I made sure using lolnexsus). All the players in my team can still chat with each other. After restarting the client, it asks me to try and reconnect to the game, but of course it doesn't work. This prevents me from playing for a few minutes, and then the client let me go on the queue again.

I'd also mention that last week, on a different PC (windows 7), every time I started a game for the first time on that day (from what I can recall), the client got stuck in the middle of loading the game. I had to close the client manually, restart it and reconnect to the game (which already began).