Well, when I got in last night I was in a testing mood so I plugged a handset into my BT line to see how the awful background noise was getting on, and the line was crystal clear. So whatever it was has been sorted out. If you've booked a BT engineer you can cancel the visit by calling 0800 800 151, so that's what I did.
This morning the cable telly engineer was due between 8:00 and 12:00, and at a couple of minutes after 11 he arrived. He studied my symptoms and nodded sagely. He plugged a little test-handset-thingy into the cable and whistled under his breath. It seems that my cable telly signal is about 5dB to 7dB stronger than it should be -- that's about 1.6 to 2 times too strong -- and he thinks that's what cooked my box. He's fitted an inline attenuator and replaced the box and now it all works. He was here about 15 minutes.
So that's all sorted. Hurrah!