I'm experiencing similar problems, so I thought I'd share my troubleshooting and experience with AppleCare. (TL;DR - None of the troubleshooting worked, engineering doesn't seem to know the cause yet, and AppleCare is sending me a new M1 Mac mini from China).
From day 2 of using the M1 Mac mini, I had problems with my Magic Mouse 2 connection. Everything would be fine for hours, then the mouse would stop responding and I'd see the Mouse Disconnected HUD message, then Mouse Connected, then Mouse Disconnected message again and it would stay disconnected until I turned the power switch on the mouse off and on.
This was the same mouse I used with 2 different MacBook Pro's and had no issues.
On my desk, the Mac mini sits about 10 inches from my Magic Keyboard (which had no problems over bluetooth with the mini), and about 18 inches from the Magic Mouse 2. This was closer than the MacBook Pro sat to either accessory.
The mouse disconnect/connect/disconnect happened on average once a day, and didn't seem to be triggered by any particular activity. I eventually erased my mini's SSD and reinstalled the OS (setup as new, not from backup) but that didn't help. I dealt with the inconvenience, until one day things went really wrong. After about 2 weeks of use, I had an incident where my keyboard and mouse disconnected and stayed that way. I tried plugging in the USB receiver dongle for a Logitech keyboard/mouse combo into the USB-C hub I had on the mini, and even that didn't work. It seemed like bluetooth and my USB ports all went dead. After a power cycle, the bluetooth devices still didn't work. I was able to get the Magic keyboard going by connecting it via USB-lightening cable directly to the USB-A port on the mini. After that, turning the Magic Mouse 2 off/on brought it back online.
Once everything was working normally again, I went into the bluetooth menu and factory reset all apple devices, then reset the bluetooth module. This was no help, because the next day the Mouse did its disconnect/reconnect/disconnect thing again.
I finally called AppleCare. The first advisor told me its a known bug in their "emerging issues" database with over 800 reports in the past 12 days. He said engineering has acknowledged it and will fix it with a future software update, but no ETA was available. I was OK with that response - just happy they seemed to know about it and have a plan to fix it. The advisor suggested that "60% of people with the issue have resolved it by disabling FileVault." I tried that, but it didn't help.
The next day, an Apple Care senior advisor called me, wanting to collect diagnostic data. He got info about my Mac and accessories, my environment (other bluetooth devices in range), and asked me to run SysDiagnose and a Bluetooth log capture when the issue happened again. This called concerned me, as it seemed to indicate that they don't have the root cause yet. The advisor wouldn't confirm or deny that. After I sent him the requested logs and diagnostics, he got back in touch with me to schedule a replacement unit for my Mac. He said engineering wanted to capture my Mac so they could look at it. Again, this (to me) means they don't know the cause or even if its hardware or software or both. I told the advisor I was happy to do the replacement, but concerned that the new unit would have the same problem since there are so many folks with this issue. He acknowledged that concern, but suggested that the new unit shipping directly from china would have an updated firmware that may resolve the issue. He was kinda vague in that explanation, but was genuinely trying to help and not just tell me what I wanted to hear.
Anyway, as of today my M1 mini is on its way to Apple, and I'm waiting on a new unit from China that isn't scheduled to ship until January 5th :(. I'll be back on my intel MacBook Pro until then. When the new unit comes in, I'll give an update on the bluetooth issue.
