This problem has plagued me for the past couple days. I noticed that one day I updated everything from the AppCenter, and it still had 2 Operating System Updates left, so I tried to update them again, it would just go from Starting to Installing, then stop and act like I had never pressed the update button.
I've tried updating from command line, running flatpak repair (and --reinstall-all), flatpak update, none of these helped.