Email notifications not working

have you checked Deliver Quietly Settings.

Have you enabled "Deliver Quietly"

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Manage notifications

To manage your notifications from the Lock screen or Notification Center:

1. Swipe left over an alert or group of alerts.

• • Tap Manage, then select from these options: Deliver Quietly: These notifications appear in the Notification Center, but don't show up on the Lock screen, play sounds, or show a banner or badge icon.
  • Turn Off: This turns off all notifications for a specific app. To turn this back on, go to Settings > Notifications, select the app, and tap Allow Notifications. 

You can also tap Settings to change your alert styles or alert settings for a specific app. 

Use notifications on your iPhone, iPad, and iPod touch - Apple Support

https://support.apple.com/en-us/HT201925