So the answer one of BlackSheep's questions is that a pair-swap was carried out.
BTOR engineers have kept trying to call me on the number associated with the metallic oath line 1. This is no earthly use as there is no phone service on this line. We are VoIP and mobile-only, and the VoIP is a different number altogether. I asked AA to fix the problem of BT having the wrong contact phone number for us stored on file, which I believe they have now done. (I don't think AA themselves were ever making the same mistake, whatever I had entered in the way of contact info on their system, they possibly were just applying a bit of common sense seeing as they can understand what it all means and what will and won't work, whereas BT don't have that knowledge.)
Does anyone know what contact info BTOR can be given to pass on to their engineers? More than one phone number as a fallback, because obviously it's no good trying to phone a dead PSTN phone or a VoIP number that relies on some particular internet connection being up, or some number that relies on the user having mains or whatever.
Can OR engineers have several numbers, including a mobile number? Or an email address or SMS? In fact, what about IM using Twitter? After all, AA do direct-tweet to me (what is it called) all the time with warnings and status updates. Twitter is free, I assume, and is platform-neutral and I assume that it is push, near-realtime IM and quite lightweight? There is a Facebook IM thing too, which Mrs Weaver uses, and that similarly is free and platform-neutral. And what about iMessage - can only dream.
Maybe some day OR might have all these other options written down too, to give them more of a chance of getting through when lots of things are broken, as they are highly likely to be.