What solutions exist when apps don't download or update on iOS 6? I've been having a problem with installing and updating apps after upgrading to iOS 6 on my iPad (3rd gen). When I click the Install or Update button the button, it turns inactive with "Installing" word on it, while the app's icon on the SpringBoard says "Waiting". 

I tried even buying a new app, but it is exactly the same. Deleting the app and trying again doesn't help. What's interesting the problem doesn't seem to affect my iPhone 4S.

I noticed several people at Apple Support Communities are affected by this, too.
I had the same problem and after none of the other suggestions worked (changing date/time, shutting down and restarting), I went to Settings > iTunes &amp; App Stores, then clicked on Apple ID > Sign Out.  After signing back in, I was able to start downloading apps again.