Because the built-in keyboard does not work even at Startup Options, this is not just a problem with the macOS login screen.
The symptoms could be related to software, firmware, or hardware, but there is not enough information yet to say which one.
The fact that it happened after the update does not by itself prove that the update caused it.
First, disconnect any unnecessary accessories and let the Mac cool down if it's unusually warm.
Then shut it down by holding the Touch ID/power button for up to 10 seconds.
Press and hold the power button again until “Loading startup options” appears, then choose Options >> Continue to enter macOS Recovery. Apple: Start up from macOS Recovery
If you can get into Recovery, do not erase the Mac.
You can use Disk Utility to check the startup disk and, if necessary, reinstall macOS from Recovery.
Just be careful not to choose an erase or restore option. Apple: Use macOS Recovery on Apple silicon
Since you do not currently have an external keyboard, borrowing a wired USB keyboard would be a worthwhile next step.
If it works in Startup Options or Recovery, that gives you useful information about the built-in keyboard.
If it doesn't work either, mention that to Apple or an authorized service provider.
Have the computer evaluated by the Professionals who have the Special Hardware Software and Expertise to diagnose the issue and offer possible solutions
Make an Appointment at the Apple Genius Bar or call 1 800 MyApple and have them evaluate the computer
Other location Call Centres
Contact Apple for support and service
If an external keyboard lets you use Startup Options, you can also run Apple Diagnostics to check for possible hardware problems. Apple: How to run Apple Diagnostics on your Mac
If the problem continues, Apple also has a firmware Revive procedure for Apple-silicon Macs.
A Revive is intended to preserve your data, but it requires another Mac and putting the affected Mac into DFU mode.
Since your built-in keyboard isn't responding, Apple or an authorized service provider may need to handle that step. Apple: How to revive or restore Mac firmware
I'd also mention the unusual warmth and the fact that the Mac restarts after being shut down when you contact Apple. If it becomes excessively hot or keeps restarting unexpectedly, stop repeatedly restarting it and have it checked.
So yes, a software or firmware problem is possible, but the symptoms don't establish the cause yet.
The good news is that there are several things you can try through Recovery and diagnostics before considering an erase, so your data doesn't need to be the first thing at risk.
