Try this.
1. Place a weather widget -the Samsung generic weather widget, on a desktop on your phone. Use it to adjust the weather update function, and just let it sit there, even if it has its own screen.
2. Add a weather widget to the Gear S2. It should mirror the data presented on the screen of the phone. If it does not, then update the PHONE's weather widget (press the small circular update symbol on the bottom corner of the phone) and see if that makes the watch update as well.
3. If that does not work then go to the weather widget on the Gear S2 and press on the first screen. This will display a more detailed weather menu. Scroll down to the bottom of that menu (using your finger or clockwise on the bezel). On the bottom of the display is an update icon -press it once or twice and your weather should manually update.
4. If that does not work then there are connectivity issues with the app. Also check to ensure the PHONE's samsung weather app has a low update period (press the widget, press "more" and "settings" and "update interval").
Good luck, sometimes this type of fiddling with the widgets gets them to work.