As I understand it, the Internet is designed to withstand damaged cables, and simply reroutes the traffic. Unless they can cut them ALL, I can't see it making a lot of difference. It's worth noting that cables fail for many reasons, and they are simply repaired by a specialised cable repair ship.
It is also worth noting that at this time of the year, defence spending is reviewed, and I suspect this has more to do with the defence industry worrying about further cuts! A good scare story often appears at this time of the year, and apart from the entertainment they give are best ignored.
The 'threat' was originally invented by a right-wing 'think tank' called "The Policy Exchange" and seems to have been got up by a conservative MP whose name I have forgotten. He seems to be non-technical and has an MBA which according to his profile is his only qualification. I imagine the think tank is reflecting the industrial sector of the UK's military-industrial complex, with the General representing the military bit.
As threats go, this one is about as convincing as a nine-bob note!