Whenever I go to the Island, the time never shows. I always stand still for a few seconds, nothing happens. However, when I do that in my town, it shows the time. How do I see the time while on the Island?
Due to the special multiplayer aspects of the island, the time displayed for all intents and purposes is your own. However, no time will appear in the game.
The answer to "Why won't the time show at the Island?" is a very boring "because it just doesn't". It isn't a bug, and there's not much you can do about it.
If you want the time to be displayed on your screen while on the island, I would offer this workaround: duct-tape a watch on your screen. :)