I got it, Nobby!
It is the IE Internet Options, which have been "fiddled with," and are now
preventing WMP from properly launching.

I fiddled with mine to where I was no longer receiving the Password Prompt
box, but still could not launch, so I went back, reset every tab's "advanced"
settings to Low, and then RESET them AGAIN to "Default." I then re-booted,
and YAY, Media Player works! BBC6 WORKS!

I did find in doing so, that I had lost Java scripting ability, which
required an install of Windows Script something or another, but then
everything was back to ship shape.

Hope this helps!

~M