I posted about this originally in the Ask a Stupid Question thread, but didn't get any responses. After a bit more digging on this site and a few others, the noise I'm hearing seems to be very similar to what other MkV/VI owners experienced as the tell-tale signs of a failing water pump bearing.
The video is pretty much the exact same sound I have, coming from the same location. It doesn't happen right away after startup, it takes a few minutes and then stays the same level and pitch until the car is shut off.
I'm taking it to the dealer today to see what they think. I'll keep you posted.