For several hours now, the "Weather" widget has not updated the data (a red exclamation mark to the left of the name), and when you try to start the widget, the message "Waiting for data" appears on the screen.

This is probably because Garmin Connect, on your phone, is not in sync with your watch, as the Weather data comes from there.

Try opening the Garmin Connect App and check if it refreshes.

If it does not, force quit it and restart it.

That was not the point at all.
I use Garmin Connect many times a day, and I also sync very often.
Just as unexpectedly as there was an error with obtaining weather data, just as suddenly it stopped.
A couple of hours ago, the Weather widget began to receive data again.

If you're on iPhone you have to set Location For garmin connect to ALWAYS in your iOS settings. 

support.garmin.com/.../

On you phone, kil Connect Mobile app, and restart it. Usually revives the weather widget.