We've had a smattering of these calls over the past year, perhaps once a month (but not from BT). Either of the flavour of "your internet will be disconnected today", or "we've detected unusual traffic from your computers". The first call we had (one of the former) was coincidentally not long after a stranger changing phone companies accidentally mis-typed / mis-quoted
our telephone number when transferring
his service to another provider. (That took ages to sort out, as neither we nor our ISP knew who was making the repeated requests, and we did get cut off when we went on holiday and didn't respond to the latest transfer letter in time.) My wife took the call, explained the bill payer (me) wasn't at home, and could she have a number to call back, please. Only afterward did she realise it was a scam.
My dad has been getting a few of these calls, although they were claiming to be calling from TalkTalk (which was his provider at the time). When the call comes through, it looks like it's coming from a UK number, but it's not a genuine number. As in the OP's case, pressing 2 transfers to an "operator" who advises that there's a been a security breach and that the internet connection will be cut off. What set alarm bells ringing right away was that there was no attempt to do any DPA checks. They then proceeded to ask how many lights were flashing on the router and what colour were they, and how many devices were connected. It was when they insisted that I go to a Windows computer to "help them resolve it" that my suspicions were confirmed.
I had a bit of time to play along, and had my Mac with me, so I sat down and listened to the guy on the other end of the line get increasingly frustrated when he was trying to explain what the Windows/Start menu looked like. It was when he asked what logo I could see and I replied that "it looks like an Apple" that the line went dead!
While I had fun winding the guy up, I can understand how people could fall for this. My dad was in the early stages of recovery from a stroke at the time and was still finding the world a bit confusing. I shudder to think what sort of a mess I'd be dealing with if it was my mum who had that sort of call.
I played that game once, when I was at home and had the free time to spend playing about. I was sat in front of a Linux machine, making for even more of a mis-match, but I fetched my wife's laptop to remind myself what menus were where under Windows so I could keep spinning out their yarn a bit. I was tickled by the way they talked me through running a 'cryptic' command to open Windows' built-in Event Viewer, and persuade me that all the normal log messages there were in fact serious faults that needed sorting out immediately(!) I kept them going until they gave me a web address to download the remote-desktop-control software they wanted me to install (so they could do whatever mischief they wanted), and then feigned confusion when the Windows executable wouldn't run on my Linux laptop. I then forwarded their phone number, the URL, etc, to my ISP, to keep their anti-fraud department busy.
On more recent phonecalls, I haven't had the time/inclination to be so accommodating to them, and tend to play awkward or contrary with them until they get fed up and hang up on me.