Iwatch

In this support article: Set up cellular on your Apple Watch - Apple Support, the following is mentioned:

• To receive SMS, MMS, or push notifications from third-party apps on your Apple Watch with cellular, your paired iPhone must be powered on and connected to Wi-Fi or cellular.

Also, see: Use your Apple Watch without your iPhone nearby - Apple Support

I believe you either need an iPhone or a cellular version off the watch. For more info, see (especially step 2):

If you can’t send or receive messages on your Apple Watch - Apple Support