Also having the exact same issue, on Gnome Web 48.3, Gnome 48 on Debian 13 (Trixie). Installed Gnome Web from the Debian packages, removed and tried the Flatpak from flathub, same results. Works in Firefox ESR, Firefox Flatpak, Google Chrome and Chrome-based alternatives.
Nextcloud is in a docker container, running version (32.0.1).
Debian updated Gnome Web to 48.5 on Trixie, and still seeing the same problem, however Gnome Web 49.1 Flatpak does not have the same issue on the same system.
I tested in Incognito and it works on 48.5. Cleared all cookies and cache via the Gnome Web settings and still seeing the issue in regular mode, but Incognito still works fine.
Same issue here on Devel Epiphany on Gnome OS. Other browsers work, Incognito works.
However, I was able to log in once to the admin account while setting up the server. I logged out to switch to my user account, and could not log in. After this first log in, I can log into no account, even after clearing everything in the privacy settings.