IOS – Google Maps’ “Nearby Transit” iOS widget doesn’t work

This appears to have been a (temporary) bug that sometimes happened shortly after an app is first configured for Notification Center. The settings indicated in the original question should be sufficient to get messages to appear in Notification Center.

As of right now, no, the new Maps app lacks transit directions. Instead, when you select Public Transit as your travel mode, it pops up a list of installed (or available from the App Store) apps that can provide local transit directions.

And, as of right now, no, there is no Google Maps app in the App Store (though you can still use Google Maps effectively from Mobile Safari).

There not much to do but conjecture as to why the Apple-made Maps app doesn't include transit. From Dan Moren's excellent iOS 6 review at MacWorld:

This is an odd solution for Apple, which usually prides itself on providing a consistent experience. But I suspect that handling public transit directions simply wasn’t an option for Apple at this point, and if it came down to offering a third-party experience versus no feature at all, Apple opted to not leave its users completely high-and-dry.