It sound to be like maybe Watchdog have confused two issues... one is fault repair costs, and the other is damage to BT equipment.
An example of the latter is where you have a tree in your garden which grows into, and damages, overhead wires belonging to BT. They will then pursue you for the costs of repairs. And that would happen regardless of whether you reported it as a line fault or, indeed, regardless of whether it even affected your line. Even if it's your neighbour's line, if your tree caused the damage (or your ladder, or your paintbrush or whatever) then you'd have to pay for the repairs.
I suspect it's damage to wires that these BT's charges may have been about (WD did mention that, but quickly swept over it), so it may have nothing at all to do with the NTE boundary of responsibility. It would then appear that, if things happened as reported, BT still behaved wrongly so there actually was some good broadcast material to be had. But that would mean that Watchdog had completely missed the point of it all.
I don't know about anybody else, but I'm getting the feeling this latest series of WD has maybe seen much intervention by the Beeb's all powerful 'dumbing-down' committee.