If you have physical access to that Mac, then you can try performing the steps outlined in the first post by @ekassos. It will be tricky to do and is best done by first experimenting with a monitor connected so you can get the timing down.
For step #2, you need to hold the power button for at least 10 seconds so it triggers One True Recovery mode (1TR). Then wait a little bit longer, maybe another 10 seconds....this is were experimenting with a monitor would help.
Now just press the "Return" key and very quickly hold down the "Shift" key (maybe you can hold the Shift key first then press the Return key, but I've never tried it). Then you wait until you think macOS has had time to reach the login screen unless you have your other Mac handy to watch for a connection.
I've had to do this a few times, but I also had the option of experimenting with a connected display first.
I'm assuming you have already tried a regular reboot to fix the issue? Maybe you can delete the macOS .plist preference file which holds the results or status of the OS update check (No idea which file or where it would be located).
FYI, be very careful using the "nvram" command on an Apple Silicon Mac since several people on these forums have reported having their Apple Silicon Macs become inoperable afterwards which then required either a firmware "Revive" or possible a firmware "Restore" (the latter will destroy all data on the internal SSD) to bring them back.
I do know that the Software Update check can get stuck like this on its own....sometimes it is due to Apple's servers being too busy. Are you using any anti-virus apps, cleaning/optimizer apps, or third party security software? If so you should uninstall them since they will interfere with the normal operation of macOS...these types of apps are not needed on a Mac.