Background App Refresh has absolutely nothing to do with notifications; notifications are send over the cellular voice channel, and when they arrive the app is launched of foreground (not background) to process them. This link explains how APNS (Apple Push Notification Service) works→If your Apple devices aren't getting Apple push notifications - Apple Support
So the first two things to verify are that you can make and receive calls and SMS text messages, and then restart your phone to fix the problem if your iPhone’s notification service has crashed.
Also, note that if you are wearing an Watch audible notifications will go to the watch, and not to the iPhone.
More Background:
Background App Refresh is confusing, and Apple doesn’t explain it well. To start with, most apps on iOS devices are “event driven”. That means that something happens that is of “interest” to an app, and the app runs to deal with the event. Examples: A mail message arrives, so the Mail app runs to slurp it up. Likewise if a text message arrives; the Messages app runs to add it to the message list. A weather update arrives, so the Weather app runs briefly to add the latest information to the weather app. The same idea for the Stocks app, Find My, Apple News, dedicated news media apps, Facebook, Snapchat, WhatsApp, etc. In addition, Notifications that you have enabled also cause apps to run to process the Notification, so you know, for example, that a new text message has arrived.
All of these examples are apps that respond to an “event” that affects the app’s content. NONE of these "event-driven" apps are involved with Background App Refresh; they actually aren’t background processes at all, the are brief executions of the app behind the screen. Background App Processing is used by apps that create their own events; for example, the Photos app periodically organizes the images in your photo library into automatically generated albums. This was not caused by any external event; the app just “decided” that it was a good time to organize albums. Or the Messages app takes time to “clean up” the message database. So it is only apps that decide, on their own, to do some “work” that use Background App Refresh. Very few apps actually take advantage of this feature that was added several years ago. If you turn off Background App Refresh the apps that use it won’t do their “housekeeping” until you open the app, which may slow down your access to the app.