I had this same issue and figured out a fix a few days ago.

Go to settings - mail -customize notifications - select mail account - sounds - choose a vibration sound and set alert tones to none.

Go back to customize notifications - remind me - choose a vibration sound and set alert tones to none.

I'm not sure what any of this has to do with receiving notifications on the Lock Screen, but it's worked for me (so far).

Also, make sure you have alerts turned on. Seems obvious, but I think iOS 16 fucked up notification settings.