If your Yahoo Wigdet gets stocked showing old weather conditions you won't find a lot of help on Google or Yahoo-help pages. So it took me long time to solve the problem including reinstall of Yahoo-widgets.
First make sure that your IE is not in "Work Offline" mode (File/Work Offline) - NO matter if you use any other NORMAL web browsers like Firefox, Opera etc. You can see youself that this thing matters, because if you try to change to any other location, you will get "No connection" message and no weather conditions will show up. So, go online with the IE.
Clean all temp and junk files on your PC
Go to Yahoo Weather Widget, open location menu (under current temperature) and Clear Current Location. You will get default location Sunnyvale, CA with current weather informatin.
Now you can select Change Location and find your favorite one (probably from the beginning of this story which got stuck) - and everything works fine.