Turning on notification in Outlook does not guarantee Windows 10 will honor them. So, you have to ensure Windows 10 notification settings are set to allow notifications from the application (they may be in the Action Center, but not displayed on the Desktop).
If you already have Outlook configured to display the desktop alert, then follow these steps to configure Windows.
- In the Action Center (in the bottom right corner, next to your time)
click on All Settings. An alternative method to open the Settings
app would be to open the Start Menu and then click on Settings.
- Click on: System.
- From the panel on the left select: Notifications & Actions.
- In the “Show notifications from these apps” section at the bottom,
Outlook. (If Outlook isn’t listed, you haven’t received a new message yet so Outlook isn’t registered yet as an app that receives
notifications. Simply send yourself a test message in that case and
reopen the Settings app.
Also note that you must be using Outlook 2013 or Outlook 2016 as Outlook 2010 and previous have their own notifications which do not
integrates with Windows 10)
- Make sure that you have enabled all notification options.
- Close Settings.
You may or may not need to restart Outlook.
Source - Not getting New Email Alerts on Windows 10
If this doesn't fix the issue, you may need to perform a repair install of Office.