I was having issues too, just upgraded to IOS 15 and I wanted a ding when a mail arrived, but no alert / notification to pop up (as I had on ios 14). I spotted the customise notifications area, and set "alert" with a sound for my email account, and when the phone was unlocked it worked (i.e I got a ding, but no alert), but when it was locked I got nothing.
Then I thought, maybe it's something to do with the Apple watch (as alerts go there when iPhone is locked), so I checked settings there. Under mail notifications it was set to "Mirror my iPhone", so I changed it to "Custom", and selected "Send to Notification Centre", now it seems to behave as I want. Whether my iPhone is locked or not, when I get a mail, the phone dings, but I don't get a alert pop up (either on iPhone or watch).
So, anyone still having issues, check your watch setting.