No call alert on Apple watch

If you have an Apple Watch with Cellular and GPS with Cellular plan active, and your iPhone isn't nearby, then you have a problem with your cellular plan.

If you have an Apple Watch with GPS, and your iPhone isn't nearby, this is expected as your iPhone uses Bluetooth to send notifications and calls to your watch.

If you have an Apple Watch with GPS, and your iPhone is nearby, then they aren't connected. Just go to Bluetooth settings and connect them.

P.S. You set the device as Apple TV :)

Try force restarting your iPhone by whatever method your version calls for.  As soon as you see the Apple logo force restart your watch by holding in the crown and side button for 10 seconds till you see the Apple logo then release.  Log into your iPhone and wait for your watch to boot, then log into it.

Are you able to make a call from your watch?

Do you use Siri on Carplay with your iPhone?