In this question I learned that you have to download patches for games on the Wii/3DS manually from the Nintendo store.
I can't think of why some games, like Legend of Zelda: Twilight Princess or Super Paper Mario would have patches, and my experience with game updates on the PS3 is that you are notified automatically about the update when you play the game (a bit too often if you ignore it).
While I have no problem having to manually download a patch from the store I wonder if the Wii or 3DS will notify you if a patch is available or not since I will rarely go into the store and a patch may not actually exists when the game is initially released (I tend to only enter the store to check on DLC)
Best Answer
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That said, 3DS games with an online component may be able to tell if you're running an outdated version, and do something appropriately.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.