Same problem her with a Macbook Air M1 2020. MacOS 12.0,.1has broken bluetooth big time.

I have done a bit more investigation myself. What I discover is that when I do a cold boot, if I let the Bluetooth mouse connect first at the log on screen and then log on, the mouse works correctly without auto reboot. However, if I log on too quickly without letting the mouse to connect first then upon log in the mouse becomes unresponsive and system will do a auto reboot. It looks like some kind of Bluetooth connection timing issue with 12.0.1. Mouse worked fine with OS 11.* . This is an annoyance. Hope Apple will fix it soon.

From this thread, I can see that that the bluetooth problem is wide spread with different devices, even with Apple devices. What I got in touch with Apple Support, all the service desk can advise is to reset/rebuild the OS. I decided not to as from years of experience, I am almost certain that this is an OS problem and not my device problem. The support person was trying to say because I use a third party (Logitech) Bluetooth mouse. My question is does anyone know if Apple Support is actively monitoring Forum. Otherwise, we could say all we want and nothing will happen.

An update from me and hope it will help other people ... I bite the bullet today and upgraded from 12.0.1 to 12.1 and my Bluetooth mouse (Logitech M557) problem is gone. My machine no longer randomly crashes when trying to connect to Bluetooth. So it appears that some Bluetooth problems have been fixed in 12.1. However some other people on other forums are still reporting Bluetooth mouse problem in 12.1 so may be the problem is still there for some people even with 12.1 I also feel vindicated that I did not take the advice from Apple Support to reinstall 12.0.1 . It would have been a huge waste of time and more frustration. Apple Support seems to just follow a standard script without real understanding of the underlying problem.