Just called BT again (just to make sure they really are f*ckwits). Again, sat on hold for a good 10-15 minutes, but got through to someone who not only had a clue, but even asked if she'd done a good job!! The moral of the story seems to be: If BT don't tell you what you want to hear, wait a few minutes and try again.
So the new line is ordered, and should be working by the time I get to the new gaff. I imagine that means the ADSL (and The Cat) will actually fall over at 9am on Thursday. I should be back up on the 13th July. Sorry for the incponvenience.