Click http://dcdudley.tripod.com/brumsayings.html
and have a look at Para 2 re Shakespeare - he being the 'Bill' concerned. I've no idea how accurate it may be but I like the concept.
I can still remember the first time I heard this phrase, and filed it away for future use. It just means the sky has turned dark ready for rain, but I just like the sound of it, so that's what I say.
This was my father's favourite expression, and was used exactly as andy hughes has described - before a storm when black clouds were looming. I used to think it was just a Dorset/Hampshire expression, but I believe it was used all over the country.