@Kalkerwelly, sounds like your neighbour will have to pay up & compensate you for the loss of your service, I assume her builders are insured against this type of issue, good luck at the small claims court. I hope Openreach can resolve this line coming in from your neighbour's property, sounds like that should have been corrected a little while ago.
BT don't care, because it's Openreach's cable. Openreach may start looking for someone to bill, though. Provided the cut wasn't made on your property, that shouldn't be you.