My provider has responded!
I've put up an explanation but I'm not actually sure whether I dare post over at The Sideshow for the moment.
Short version - I had actually been exceeding bandwidth for a couple of months but they suddenly cut me off without warning, claiming the problem had arisin "in the last 24 hours", which obvously wasn't true. I suppose this is my fault for not noticing what my usage limit was to begin with. That's just the sort of thing that would escape my notice. But it does have to be said that my provider isn't particularly good at making things clear. *sigh*
So they want me to upgrade and I've asked them to give me appropriate details and in the meantime I'm comparison-shopping and looking into getting The Sideshow it's own domain name (one that doesn't involve frames) so further emergencies (or future migrations) don't result in this sort of thing again.