Apple Watch "Not Delivered"

On your iPhone, go to Settings > Messages and make sure that iMessage is turned on. Then tap Send & Receive and make sure that you're using the same Apple ID that your Apple Watch is using. If you aren't signed in, sign in to iMessage with your Apple ID.

https://support.apple.com/en-us/HT208489

If this relates strictly to traditional text messaging (SMS) it is likely the problem is due to your watch not getting a delivery confirmation back from the network. SMS has two stages in each direction, you to network, and network to recipient, so there are two places a failure could occur. As it is every message it is unlikely to be at the recipient side because they would not all have the same problem sending the acknowledgement, so the most likely problem is from your network back to your watch.

Questions to help focus the problem:

• Is your watch the cellular version used on standalone cellular, or the GPS (or cellular working over Bluetooth through your iPhone.
• I have only considered SMS above. Does the problem also impact iMessages to other iDevices?
• Do messages sent from your iPhone show the same failure?
• How soon after sending do the fail reports appear?
• Did you check with your network for any ongoing service issues in your area which might be causing problems?

The lack of problems with other devices tends to point at a failure between the iPhone and the watch. There is a very high probability that lack of Bluetooth (the default connection) could cause the problem. Fortunately this will be easy to test by activating Bluetooth.