Just to finish this off, I've closed the BT anywhere account and happily signed up with a sim-only deal.
I had no idea that I was such a dinosaur in still having a BT mobile deal. I spoke to quite a few staff, as it took several attempts to get the whole thing closed down - I wanted to take the broadband off the line, but keep the phone line, which for some reason was complicated. A couple of them mentioned how there aren't many anywhere customers anymore.
My remaining task is to retain a btinternet email account that disappears as the broadband closes. I'm told that they will try to charge me for it if I go online, but I can bypass that by phoning and getting it transferred to the infinity account I have on another line.