Nintendo of America's product marketing manager (and often Shigeru Miyamoto's translator) Bill Trinen has said users will need both machines hooked up to an internet connection simultaneously to transfer over your account and associated purchases.
In a process apparently similar to 3DS-to-3DS XL transfers, Trinen told 1Up, "Essentially you'll have your Wii and you'll have your Wii U. Those two will connect and the data transfer will happen and you'll transfer the SD card over. Then you'll have access to the digital content that you downloaded for Wii and you can play that on Wii U."

He goes on to clarify, "And of course all the Wii software itself will be backwards compatible."

So if you've bought a load of digital games on your Wii, you'd best hold off trading it in until you've done the whole transfer business.