The problem seems to be caused by an update to SDDM (Simple Desktop Display Manager).
On Fedora it can be downgraded, but first you need an internet connection. I enabled my wireless internet connection by running:
nmcli connection up <wireless-network>
<wireless-network> is the SSID of my wireless network.
I then downgraded SDDM:
sudo dnf downgrade sddm
Logins should then work after a reboot.