I just got my Apple Watch series 4 on Thursday and got it set up. If my phone is off I can only receive messages from people with iPhones and phone calls. I have checked with AT&T and the cellular is setup correctly on their end. We have tried unpairing and repairing the watch twice. And it still won’t work.
If you are referring to a regular SMS, not an iMessage, then see this, How to use your Apple Watch without your iPhone nearby - Apple Support
To receive SMS from third-party apps, MMS from third-party apps, or push notifications from third-party apps on your cellular model of 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.
If you are referring to a regular SMS, not an iMessage, then see this, How to use your Apple Watch without your iPhone nearby - Apple Support
To receive SMS from third-party apps, MMS from third-party apps, or push notifications from third-party apps on your cellular model of 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.
Hi there:
I have an Apple Watch Series 4 Cellular and my paired Phone is powered on and connected to WiFi or Cellular, but I do not get third-party apps push notifications.
Thanks
My watch won’t recieve sms messages
