I have not had reliable notifications in either mail or the message app since purchasing my iphone 12, 2 years ago. At times they will work, mostly they don't. But it seems that even if they start working, every time Apple comes out with an upgrade to ios, one or the other doesn't work again.
Currently, (ios 15.5) my messages app notifies me, the Apple mail app does not, but Gmail does so I do have it installed for that purpose.
It's actually rather comical but pathetic that I have to use the competition's email client to get my iphone to work the way it's supposed to.