Hey, someone actually posted a similar problem; to get the patch; follow the instructions below:
Nintendo Servers: You've said your game wasn't patched, which means your Wii hasn't downloaded the correct patch to connect to the COD servers after the official Nintendo Wi-Fi shutdown. To download this patch, make sure your Wii is connected to the internet, you have enough space on your Wii to download a relatively medium/large size patch, and that the region of the game you're running is the same as your Wii (PAL game= PAL Wii; NTSC game= NTSC Wii). If these requirements are met, just connect to multiplayer > Nintendo Wifi and a prompt should come up asking you to download a patch. Accept the prompts and upon returning to the multiplayer section; the selection screen should say "MW3 Online" rather than "Nintendo Wi-fi"
Hope this helped. Please post updates
EDIT: The patch will actually save to your Wii, so be careful while messing with your save files.