I have macbook air os monterey and first blue tooth did could not switched on , after downloading new Os Montery bluetooth is on but does not connect mouse. pls help

If all else fails give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Re: "after downloading new Os Montery bluetooth is on but does not connect mouse. pls help"

Apple support have articles online that may help:

How to: Set up your Magic Keyboard, Magic Mouse, or Magic Trackpad with your Mac - Apple Support

More info: For Monterey: How to: Connect an Apple wireless accessory with Mac - Apple Support

How to: Use an Apple wireless keyboard, mouse, or trackpad with Mac - Apple Support

And

What to do:

If you can't connect your Magic Keyboard, Magic Mouse, or Magic Trackpad to your Mac - Apple Support