"My gran has moved house, and they say they need to send an engineer round, which will take 3 weeks and cost £125!"
Argue.
When I moved home last year, I read the bit on BT's web site that reckoned that if there was an existing line at the new address there would be no charge. Knowing that the previous guy had cable, so the BT line here was disconected, ancient and probably naff, I specifically asked them whether this still applied in the circumstances, and was assured that it did.
As expected, an engineer visit was necessary. Even with the exchange being just at the end of the road, what with trotting down there for testing, running a new line from the pole, fitting a new master socket etc., it was about an hour's work or so.
They then stuck £100+ on the first bill that I got here - but when I reminded them of what they had said prior to the move, they did refund it, promptly and without any shilly-shallying.