Hello JamesBarr4,
Thank you for using Apple Support Communities! It sounds like notifications for the YouTube app on your iPhone aren't working as expected. You're in the right place to find a resolution.
Following the steps in the resource below, let's start by checking your notification settings for this specific app.
Use notifications on your iPhone, iPad, and iPod touch - Apple Support
You'll want to ensure in particular that YouTube notifications aren't turned off, set to Deliver Quietly, and that you have sounds and alerts set. Keep us posted with what happens after trying those steps.
Cheers!