The scam you have posted above has been happening for a number of years now and is in fact mentioned in this thread along with the method of how it is perpetrated. It is also posted in other threads on the forum.
While it is not a new scam it is always a good thing to alert other users that the scam is still doing the rounds.
Always good to have an update.
I got a call from these people today with exactly the same scenario - from an 01256 area code number. I've had a few more in the last few weeks all from the same area (but with different phone numbers). Looks like the people on Basingstoke have nothing better to do.
Apart from the ladies attitude and accent it was pretty well put together scam. I called BT after, they don't call they wil email any issues.
They don't seem to be interested in following up.
I had a call a few weeks ago that ended abruptly when the lady called me a liar when I told her I didn't have BT broadband. Definitely not BT at that point. Pot, kettle, black...