Why aren't third-party app notifications working on iPhone 13 Pro with iOS 16.3.2?

Check this, first.

Or Use and customize the Action button on iPhone - Apple Support (IN)

Then please check the following.

• Check your DND status - Settings > Focus [Do Not Disturb]; Use Focus on your iPhone, iPad, or iPod touch - Apple Support
• Check BedTime Status - Clock App; Use Bedtime to track your sleep on your iPhone - Apple Support
• Check your iPhone Sound Volume; Adjust the volume on your iPhone - Apple Support
• Check Call Audio Routing; Call Audio Routing Bluetooth / Speaker / Automatic
• Check Auto Answer Calls; Turn off Auto Answer Calls
• Check the list of Blocked Numbers --> Block phone numbers, contacts, and emails on your iPhone ...

Notification Preview when Screen is Locked, Notification Alerts, Notification OFF, and much more...

1. To get Notifications when the Screen is locked see the pic below which is self-explanatory
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2. To turn ON notifications selectively for apps, go to Settings > Notifications > Scroll Down to the App and turn ON notifications... Change notification settings on iPhone - Apple Support (IN)
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3. Some apps have their own notification settings over and above the iOS notification settings: An example of Instagram Notification settings is given below
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4. Help for Notification Alerts --> No Notification Alerts - Apple Community