Yesterday evening, I re-installed Steam, as I already have an account and some bought games (I just haven't opened them in a few months).
Then, I left my computer turned on over night, just for downloading Team Fortress 2, as I want to play it tomorrow evening.
This morning, 80% of it were local (around 5Gb if I remember correctly).
Then, Steam asked me if I wanted to install an update from Steam; I chose yes, restarted Steam, and my download progress reverted to zero.
The only way I could find to make Steam download anything again was to remove all local game content and start over.
I just hope it gets here in time for tomorrow afternoon.