Does a Wii/3DS automatically notify you of game updates?

In general, no, neither the Wii nor the 3DS is aware of whether an update or patch is available. You have to pay attention to Nintendo news online to be informed.

That said, 3DS games with an online component may be able to tell if you're running an outdated version, and do something appropriately. For example, Mario Kart 7 will error-code you if you try to race online without updating to 1.1. This probably depends on server-side code so a Wii could feasibly do the same, though I don't think such a game exists.

If you're lucky, Nintendo may send out a message to all systems telling people about the bug and/or fix, like how they did for the Pokémon X/Y save location freeze bug.