The Last Mellon

Online will work with your friendcodes for those games.
It might generate a new code unless Nintendo figure out a way to get your codes from your old Wii.
That might be unlikely, it depends what they're going to do.
I hear they plan to bring your Wii software over via Club Nintendo, but saves will probably just be you copy them to SD Card.

Mario Kart and most other online gamesaves are locked to your wii, because of the friendcode system, but Nintendo could unlock them for transfer if they wanted to.
At the same time, would they bother? Most people would be fine with starting from scratch so long as it's just their online games that won't copy over.
It wouldn't kill them to make an app for it, there's a homebrew that does it, but at the same time, they never bothered to update the Wii's Mii plaza to allow you to transfer 3DS Mii's over to your Wii did they?