Hello, same problem happens to me.
The download of the package was ok.
When hitting (4 or 5 times) the RESTART button inside of the updater app nothing happened.
I closed the updater, reopened, downloaded the OS update and hit restart. Went fine.
The phone rebooted, I left it apart.
After 30 min I checked the state of the update and found the phone stocked [at a screen like this one] which appears to be the TWRP (https://forum.fairphone.com/uploads/default/original/2X/0/02fa5f3a67356e8f746e85b13356b398f3e8892d.png) .
If I reboot the phone is stocked at the very first loading screen (FAIRPHONE POWERED BY android).