Measurement of the power passing through some cables is simple stuff, even though the new metres doesn't contain moving parts, there is no radically new measurement technology in "smart" metres as compared with the old metres. Reporting the electricity consumption in real-time is done through cellular networks, so it shouldn't cause any different radiation than ordinary cellphones.

The term "nuclear radiation" generally covers three kind of radiation, there is the alpha rays (high-speed atomic nuclei particles), beta rays (high-speed electron particles) and gamma rays (high-energy photons or high-frequency electromagnetic waves). The simplest way to generate such radiation is through embedding some radioactive components (i.e. "nuclear waste") in the product, like they do with smoke detectors - but I'm fairly convinced they don't put radioactive waste into those smart metres.

Apart from that you would need really heavy particle accelerator to create alpha radiation. To generate beta radiation one could do with a fairly smaller particle accelerator - like, the cathode ray tubes (CRT) used in old TVs and computer screens - though, the electrons aren't accelerated nearly fast enough for them to be dangerous - and I'm fairly certain there are no CRTs inside the smart metres.

"Gamma rays" is electromagnetic radiation, but as far as I know there are no way to produce such radiation electronically.

Microwaves are also electromagnetic radiation.

Whatever apparatus was used for measuring the radiation, I believe it would have been making the same (or higher) readings if exposed for a cell phone during a telephone conversation.