I think this may be related to the dialog problem described here.
Standard dialogs are not working at all, such as alert, confirm or prompt.
The login prompt that is shown to authenticate the user is probably blocked (does not work or is not visible) and that is why the web app does not pass through the authentication phase.
I suppose Apple will have to fix this bug in a future release.
Edit: After upgrading to iOS 7.0.3 basic authentication suddenly started to work again also in home screen web app mode. Login prompt is displayed and everything works as expected.