Did some testing. Email notifications are not appearing on the lock screen, either.
Here are my settings (for two email accounts):
- Alert style: Banners
- Badge App icon: On
- Alert sound: <On>
- Alerts: Show in Notification Center: On; Show on Lock Screen: On; Show Preview: On
So, none of these settings seem to be working.
Mine work fine, both sound, as well as banner notification and Lock screen preview.
When you mentioned that you reset the phone - was that using "Reset All Settings"?
Are you by any chance using any VIP addresses?
I remember back on iOS 6, when I entered some VIP addresses, I stopped getting email banner notifications, until I deleted those contacts from my VIP list.
Reset All Settings does not wipe the phone. Erase All Content and Settings will wipe the phone.
To be clear, using the Reset All Settings option will not erase apps, will not erase photos or music, and will not erase email.
It will simply set all settings back to "stock" settings.
Ringtones and notifications will be default settings.
Wallpaper, brightness will also be default settings.
Of course you will have to reenter WIFI logon settings, and reset wallpaper.
Best of luck!
If you are afraid of losing anything, you can always backup to iTunes first.
Then Reset All Settings or even Erase All Content and Settings to see if the device malfunctions as STOCK.
After that (once you've done all your testing), you can restore the iTunes backup back in.
It's always nice to do this before a Genius Bar appt so that they will know if the product malfunctions "out of the box" so to speak.