Apple Watch series 3 will not update. It says it is not connected to the internet
This watch will not update. It's a battle every time to get it to update.
Apple Watch
When the watch is on the charger, check that the Wifi connection is good (under Settings). Check also progress under Settings > General > Software Update there. That can tell you more clearly than your iphone what step is going on at the time. You may also need to enter passcode there for the update to proceed. This appears to be typical for S3.