Hi RolandMetikos,
Thanks for coming Apple Support Communities for help today.
It's possible your mail provider doesn't support Push.
Let's try restarting your iPhone first as this can often resolve minor issues. You can also do this after toggling notifications off. Once you have restarted toggle them back on and see if it triggers them into working again.
Also be sure you have updated to iOS 16.0.3 to be sure being out-of-date isn't related to the issue you are experiencing with notifications.
If the issue is still occurring, make sure the issue happens on both Wi-Fi and cellular data to further isolate the issue.
Let us know what happens.
Have a fantastic day!