A search in the documentation finds several pages about backups, this here among them … https://doc.e.foundation/support-topics/backup-and-restore-ephone.html
Installing manually can be done with the Easy Installer or via command line … https://doc.e.foundation/devices/FP3/official … both methods will wipe user data.
But since you are on Android 10 (Q) and stable, I would perhaps wait if something comes up to either solve this failure to upgrade or at least explain it.
At least the OTA server still knows about those old versions, so it should be able to offer something to the updater.
Are you on a Wi-Fi internet connection when you try to find updates?
Do you use any Antivirus or Firewall App you could deactivate to see whether they were blocking the updater?
@Manoj: 0.12-q-2020111084009-stable-FP3 should still be OTA upgradable in theory, right?
If nothing helpful comes up otherwise … There is a method to update in recovery mode to a newer version within the same underlying Android version without wiping data, in effect emulating what the updater should do, but this isn’t officially recommended or documented, and backing up everything important from the phone to somewhere safe beforehand should be done before trying this, too.
I don’t know whether everything was in place for this already with the 0.12 recovery, but if you want to try and find out and have no fear of the command line, perhaps open a new topic to be walked through this.
The idea would be to get to 1.4-q-20220920220045-stable-FP3 this way and see whether the updater can upgrade by itself from there.