I got my internet connection back today.
There was a BT engineer going up telephone poles in the area on Wednesday, and our telephone line went dead at some point in the early evening, while our ADSL connection still worked but was running at about 36Kbps on what should be a 5Mbps line. On Friday, another BT engineer came to fix whatever the first one had fucked up, and he got the phone system working, but the ADSL was still slow and was running at 56Kbps. We kept calling the technical support line, each time having to go through the standard quizz of what our phone number was, what model of router we used, whether our test socket had a BT logo on it etc, administered by people in an Indian call centre with accents that made it difficult to understand what they were saying, and each time were told one of several stories; that there was a problem with our line and they were working on it, that there was a problem with our line and we needed to contact Talk Talk (we have different companies supplying our telephone and broadband services; Talk Talk supplies the telephone service, BT supplies the broadband service), or that our line was fine and the problem was with our internal wiring or settings.
On Saturday, the ADSL stopped working completely, but now it's back and the speed is normal again. Unfortunately, I think BT are going to be replacing a buried cable up the street soon, so we might have to go through this process all over again. Also, the connection still seems flaky; random sites keep timing out for no obvious reason when I try to contact them.