I'm having the same problem too. Before watch OS2, it was the sunrise/sunset which would go blank but weather at wherever location I was at was consistently good.
Now after updating to WatchOS 2 and on the iPhone 6S, the location weather seems to have stayed on the location it had locked on earlier today even though I'm now at a different place. Have tried shutting down both the weather app on the watch and the phone but once I launch the app on the watch, it keeps going back to the earlier location.
When I turn off the location services for the weather app and back on (on the phone) the watch then keeps showing the loading graphic symbol, with "local weather" at the top of the watch face but nothing loads. It will spin endlessly with no progress and I end up closing the app.
I've had to shut the watch down completely and reboot it. Then when I check the weather it will say "Cannot access location data for weather" even though the weather app on the phone is displaying the location weather fine. After a few minutes, the watch will suddenly decide it will load the location weather after all
However when I moved to a different area and rechecked the weather for the new location I was at, the watch was now back to giving me the weather for the previous location and had not updated to the new location.
What an annoying bug! I don't want to have to shut down my watch overtime I want to check the local temperature!