Same thing happening to me. I've got a Gear 2, which runs off that weather widget's data, so the watch doesn't update either. Even if I push a refresh from the watch, it won't update unless I unlock the phone. For me the weather will update if the phone is unlocked, but doesn't update at all on its own no matter what the refresh rate is set to. The watch works fine otherwise, I get email notifications and call notifications correctly, so I know their communication is working and that the phone is still communicating with the rest of the world when the screen is off. Seems like it might be a bug with the Accuweather widget itself.
I have location services on, but Data Roaming is off. Perhaps that's the issue, but not a compromise I'm willing to make just yet. However, 90% of the day the phone is on a good wifi connection, so it shouldn't need to be roaming around cell networks to update.