Apple hasn't commented on the issue, but some smart users seem to have already figured out a possible explanation of the problem, and the issue has been fixed by Apple. The theory is that since iOS 4, the platform standard is to ping a page on Apple's servers: http://www.apple.com/library/test/success.html. If the ping was successful, the WiFi was deemed connected and everything would be fine. If the ping failed, the system assumed you were on a network that required a login and would ask for the credentials.
Apparently, someone at Apple accidentally deleted that page from the server, so pings were failing all around, and no one could connect to any WiFi. The page has since been restored, and no one is reporting any more issues, so it seems that really was the problem.
Anyone else having issues with iOS 6? What are your early impressions?