Description:

Detailed instructions on performing a system transfer between a Wii console and a Wii U console

Important Notes!
Make sure your Wii U console has the latest system update (How To).
If your Wii console is currently locked with Parental Controls, ensure that you are able to unlock it. If you have forgotten the PIN, you will need to reset it before you can proceed. Click here if you have forgotten your parental controls PIN.
Software and save data that may already exist on an SD card from the source Wii console cannot be transferred directly. If you have data on the SD card, perform the following:
For software, either transfer it back to the source Wii console before performing the transfer, or redownload it after the transfer using the Wii Shop Channel on the target Wii U console's Wii Menu. (How To)
The following video will give you a helpful overview of the system transfer process:

What You Need:

Your Wii and Wii U console (at least one Wii Remote or Wii Remote Plus is required to control the two consoles)
A SD or SDHC card with at least 512 MB of free space
A broadband Internet connection (both systems must be connected to the internet)
A television
The country settings on both consoles must match
What you can and can't transfer
The following is tranferred during the system transfer process between a Wii and a Wii U:

Wii Software Save Data
WiiWare and Save Data
Virtual Console titles and Save Data
Add-on Content
Wii Points
These points will be combined with the points on the target Wii U system, but cannot exceed a combined point total that would exceed 10,000 Wii Points
Wii Shop Channel Account Activity
Mii Characters
Nintendo Wi-Fi Connection data

The following is NOT transferred during the system transfer process between a Wii and a Wii U:
Data from Wii settings
Preinstalled WiiWare software and preinstalled Virtual Console titles
Software and save data for which transfer is not permitted, such as WiiConnect24 data
Software that is present on both consoles
While the software will not overwrite what is already on the target Wii U system, the new save data will overwrite what is on the Wii U
Nintendo GameCube Game Disc save data
The following channels, games, and applications cannot be transferred:
LostWinds
Internet Channel
Everybody Votes Channel
Check Mii Out Channel
Nintendo Channel
Metroid Prime 3 Preview
Photo Channel 1.1
Netflix
Hulu Plus
Wii U Transfer Tool
YouTube Channel

Additional Information:
All data that can be transferred will be transferred at one time. You cannot select what data is transferred.
Any software or save data that exists on an SD Card from the source Wii console cannot be transferred directly.
For save data, transfer it from the SD Card to the source Wii console before performing the transfer.
For software, either transfer it back to the source Wii console before performing the transfer, or redownload it after the transfer using the Wii Shop Channel on the target Wii U console's Wii Menu.
Mii characters are moved to the Mii Channel, accessed from the Wii Menu application on the Wii U console.
WiiConnect24 functionality is not supported on the Wii U. You will not be able to use features that require WiiConnect24 after the transfer.
Depending on the software, some features may be limited after the transfer.
Important Note!

If both systems are not connected to a television, you will need to switch the television connection between systems in order to complete a system transfer.
You can either:
Use the input select of the TV to switch between the two systems; or
Plug the standard AV cable that came with your Wii console into a TV and swap the cable between the two systems during the transfer process. (Swap the cables at the same time you are swapping the SD card.)
Transferred data is permanently deleted from the Wii console. Once you have transferred data to the Wii U console, it cannot be transferred back to a Wii console.
Data cannot be transferred from your Wii U console to a Wii console.
For SD card troubleshooting, click here.
For Internet connection troubleshooting, click here.

Setup:

Connect the Wii to a television.
Connect the Wii U to a television.
If you only have a single television, you will need to connect both systems to it.
If your television has both HDMI and Audio/Video inputs, then connect both systems to the television and switch inputs when appropriate.
If your television only has Audio/Video inputs, then you will need to switch the AV Multi Out Plug on your Wii AV Cable or Wii Component Cable between consoles when appropriate.
Connect the Wii to a broadband Internet connection.
Connect the Wii U to a broadband Internet connection.
Download the Wii System Transfer tool on the Wii console. You can download the tool to the system memory or an SD card; however, if you download it to the SD card, you will need to launch it from the SD card menu.
Download the Wii System Transfer tool on the Wii U console by clicking the transfer tool icon on the Wii Menu and following the on-screen instructions.
Although there is an icon on the Wii Menu for the Wii System Transfer tool, it is not the actual tool. Clicking it will redirect you to download the actual tool from the Wii Shop Channel. Once the download is complete, the actual tool will appear as a new channel on the Wii Menu.
You can download the tool to the system memory or an SD card; however, if you download it to the SD card, you will need to launch it from the SD card menu.

Begin with the Wii U:

Insert the SD card into the Wii U console.
On the Wii U console, start the Wii Menu application and select the Wii System Transfer tool that you previously downloaded.
You will need to sync a Wii Remote to the Wii U console in order to proceed.
If the Wii System Transfer tool is not available in the Wii Menu application, perform the latest System Update.
Follow the on-screen instructions, making sure to read carefully through each before you proceed.
Remove the SD card from the Wii U console and insert the same SD card into the Wii console.

Connect on the Wii:

With the SD card inserted in the Wii console, launch the transfer application to begin retrieving data.
You will need to sync a Wii Remote to the Wii console in order to proceed.
Follow the on-screen instructions, making sure to read carefully through each before you proceed.
When the data transfer is complete, remove the SD card from the Wii console and insert it into the Wii U console.

Finish on the Wii U:

Confirm the message and select "Transfer."
Follow the on-screen instructions, making sure to read carefully through each before you proceed.
The data is saved on to the Wii U console and deleted from the SD card.