Are you getting any sort of error code? I know in the past there were issues with the Wii U YouTube app (and the 3DS YouTube app...and the Wii YouTube App), but at least some of those were addressed through app updates. Does the problem happen if you watch YouTube through your browser? I know that is not an ideal solution, but it might be a workaround that spares you from hours of frustration. Other than that, the only thing I can think of off the top of my head is to delete and reinstall the app. Sorry, I wish I could be of more help.
Nintendo of America generally has pretty good technical support, although of course how knowledgeable of a person you get on a call is luck of the draw. If nothing else works, give them a ring at 1-800-255-3700.