On Galaxy games won't update automatically, they will just appear in Downloads and you can click them to update (I think this is because the update is scheduled?). Is there a way to set it that updates just automatically download?

Settings > Game features > Game data features > [✓] Auto-update games

I already had that ticked.

The only schedule you can set up is a bandwidth limit during certain hours. It's not possible to cap the speed at 0, so this setting shouldn't matter in this case. You can see if unchecking "Pause downloads when playing" makes a difference. That's under Cogwheel - Settings - Installing, updating as well.

Also check your individual game settings under "More" button (next to Play button) - Manage Installation - Configure - Installation. Double-check that auto-updates isn't unchecked there.