Hey all, I'm having this same issue now as well. I'm very new to this all however, so a lot of what I've read here is all Greek to me and I'm having a little trouble understanding it all.
What I'm most unsure about though is that everything was working fine up until just 20 minutes ago. I booted up Need For Speed: Most Wanted and noticed it hadn't saved my progress (again, I first noticed it this morning but just assumed I had forgotten to save), so I booted up Baldur's Gate: Dark Alliance instead and had what the OP described happen. Now it seem that any GameCube game I try booting just shuts the Wii U down after a few seconds.
I did have another issue last night, however, where after I installed Rock Band 1, 2 and 3 my HDD started making a click sound and wouldn't load anything. I at first panicked until I unplugged everything, plugged it back and that seemed to have solved it. I'm guessing this might related somehow? TIA