Question: My bluetooth mouse batteries died, and when I replaced them, the mouse didn't reconnect. Now how can I get OS X to discover and connect my mouse without having a mouse to open the mouse dialogue?

Answer: 
Make sure your mouse has been reset.

1. Press `⌘`+`Space` and type "Bluetooth" to open bluetooth in System Preferences. You will see your mouse in the list of devices and a button to 'Connect'.
2. Use the `Tab` key to move to the next control, this will highlight the next one for you (If you are stuck on the text field, press `Ctrl`+`Fn`+`F7` to switch between navigation of all controls on the screen, press `Ctrl`+`F7` to switch back).
3. Once nothing is highlighted in the window, press the `Space` bar, this in turn selects the 'Connect' button for you.
I can usually get my mouse to connect if I can get to the bluetooth preferences.

Since upgrading (now on Monterey 12.3.1) that has become more difficult and a lot of the keyboard combinations to get to the menus do not seem to work for me any more.

The ⌘+Space "Bluetooth" to get to the preferences was not working until I found that "System Preferences" has its own option under Spotlight Search Results and it was not enabled. Enabling that has made the preferences show up in Spotlight so I should be able to nudge Bluetooth to connect the mouse next time.