I upgraded to a cellular S5 but I couldn’t get cellular setup...kept getting cryptic messages that I had to have it reviewed at an AT&T store. Took my phone and watch in and they sorted things out and got my watch up and running. As a test, the rep had me power off my phone. She then called my phone’s cell number and my watch rang and I answered...So the phone can be completely off.
The only thing that won’t work are notifications with the phone off...from apples support doc...
To receive SMS, MMS, or push notifications from third-party apps on your cellular Apple Watch, your paired iPhone must be powered on and connected to Wi-Fi or cellular, but it doesn't need to be nearby. You also need to be
signed in to iMessage on your iPhone.