So if apple did the same setting checks as I did it may be as well to go through the standard progressive diagnostics below.
First of all I need you to accept that no notifications will be received if the iPhone is unlocked i.e. with the screen lit up.
Standard Diagnostics:
If all settings normal but behaviour is irrational there is a progressive hierarchy of remedies. Hopefully you will not need to go to the end of the list.
First force restart the Watch
i.e. press and hold both the side button and Digital Crown for at least 10 seconds, then release both buttons when you see the Apple logo.
Then, although it may seem irrelevant, while it is restarting Force Restart the paired iphone Force restart iPhone – Apple Support (UK)
You may also want to make sure the software on the watch and iphone are both up to date to the same level. Updates are in pairs.
Update your iPhone or iPad - Apple Support
Update your Apple Watch - Apple Support
Test it for a while. If it is still misbehaving... Unpair and set up again
Unpair and erase your Apple Watch – Apple Support (UK)
and then set up without restoring the backup
Set up your Apple Watch – Apple Support (UK)
Make sure you are not using VPN or security apps on the iPhone.
If you've completed that troubleshooting and the issue is still not resolved, Contact Support again.
