Flatpak updates not found
Software is reporting that no new updates are available, but running
flatpak update and
sudo dnf upgrade on the commandline shows that updates are indeed available. The problem of no package-based updates is probably the old issue of PackageKit and dnf using different caches, but the problem of no Flatpak updates being found by Software is likely a separate issue.
I have enabled both Automatic Updates and Automatic Update Notifications in Software's Update Preferences, so I would have expected Software to have noticed the available updates. The "last checked" time on the Updates page is from several hours ago. Even when clicking the refresh button on the Updates page, no updates are found, not even Flatpak updates.