The built-in cellular feature on Apple Watch Series 3 (GPS + Cellular) requires a carrier plan for your iPhone that specifically includes support for the Apple Watch, including the carrier activating the device on the plan:
Once enabled, the cellular capability can enable certain functions to remain available on your Apple Watch when it is way from the paired iPhone and when those functions would not otherwise be available.
It may help to follow the instructions here, including under the heading "Get help if you can't send or receive messages" and including unpairing and re-pairing your Apple Watch if necessary:
Keep in mind that the functions that are available on your Apple Watch when it is away from your iPhone depend on various factors including:
- Your Apple Watch model:
- The features of your carrier service plan:
- The type of connectivity that is available to your Apple Watch at the time:
Features and requirements that apply under various circumstances are detailed here, including within footnote #1:
If you still need help, then I suggest that you contact your carrier in the first instance.
If the issue persists, then, depending on your carrier's advice, I suggest that you next contact Apple Support (remote diagnostics and/or mail-in service may be available, if required), make a Genius Bar reservation or visit an Apple Authorised Service Provider for assistance: