Well, the article at the BBC was written by someone with only a little knowledge of nuclear weapons, Electro-Magnetic Pulse (EMP) or Coronal Mass Ejections (CME). It pretty much reduces to "something bad could happen and somebody ought to do something about it." (Which is typical of FAS scaremongering.)

That said ...

Nuclear weapons could be used to generate a High-altitude Electro-Magnetic Pulse (HEMP) that would destroy or degrade a lot of electrical and electronic systems. How much and how widespread depends on the design and installation of the electrical/electronic device/system and the structure of the attack. The kind of damage the good MP is concerned about would take a pretty sophisticated attack. The yields and launch systems required rule out all but the US, Russia, and China. (Possibly the UK and France could be included.) The idea that a rogue state, such as N. Korea or Iran, could do that is ... bizarre.

CME is a serious threat, largely because the standard approaches to protecting electrical systems from everyday events, such as surges and lightening strikes, would be turned into a vulnerability. However, with sufficient warning, which we would have, the damage can be minimized by simply shutting down the power grid for a few to tens of hours. (Look up the Carrington Event, which was a CME striking the Earth.)