Since, as 'we' the community not sure about BT yet, know that there is a problem with the Exchange in Dalton-in-Furness.
A mod arranged an engineer to visit my house this friday (24th Jan). Should I just cancel it since he is not going to solve any problems for me and to save both of us's time?
i would let him still come, while he can't fix the exchange fault he might find problems with your line and repair them which will always stand you in good stead in the future.
The only problem would be if he found a fault you were unaware of in your house you could be billed for the visit.