Welcome to iMore! There are a few things you'll want to check:
1. Go to Settings > Notifications > Mail (or whatever third party mail app you're using) and make sure that ALLOW NOTIFICATIONS is toggled ON for all listed email accounts.
2. Click on each email account and set the individual alerts for that account (i.e. show in notification center, sounds, app badge, alert style, etc.).
3. If you're still experiencing issues, soft reset your phone by holding down the power button and home button simultaneously until you see the Apple boot logo.
Hopefully that fixes your issue. If you're still having problems post back and we can troubleshoot additional steps.