Having a web interface isn't necessarily enough, it has to actually stream the media over HTTP (in MP4 format) for it to work. I think, anyway. Just going off the stuff said in this thread.I do believe Universal Media Server has a web front that could be compatible. Anyone using a ps3 has probably familiarized themselves with the interface already.
I don't know if this is the case for UMS or not (downloading it right now to try and see), if so then it could potentially already work, negating the need for a specific "WiiU Media Server".
EDIT: I've been reading up a bit more about streaming content over HTTP; by the looks of things, it may not be possible to do it with on-the-fly encoding. It may be necessary to encode everything prior to streaming. Although, I'm not completely sure as there seems to be a lot of contradictory information around the internet, and talk of various protocols for streaming over HTTP, I'm not sure exactly what the WiiU uses. If anyone does know, would it be "HTTP Live Streaming"?