It appears that there is something going wrong with YouTube at the moment. A lot of people across the web are reporting that they are unable to play YouTube videos on the Wii U browser. It has been confirmed to be a glitch effecting some users but not all. Many people are reporting no issues while others are getting their pitchforks ready.
You can still use the YouTube application on the Wii U, but it isn’t nearly as good as using the video content giant through the browser.
I haven’t tested this myself but so many people are having this issue that it looks like its widespread enough to be a concern. Test it yourself to see. Perhaps you’re one of the lucky people unharmed by this glitch. I imagine Google will have it up and running in no time.