Wizarding radios are magically modified.
...many radios have been legally modified by the wizarding community for their own use, which broadcast regular wizarding programmes. The Ministry conceded that Muggles frequently catch snippets of advice on, for instance, how to prune a Venomous Tentacula, or how best to remove gnomes from a cabbage bed, but argued that the radio-listening Muggle population seems altogether more tolerant, gullible, or less convinced of their own good sense, than Muggle TV viewers. Reasons for this anomaly are examined at length in Professor Mordicus Egg’s The Philosophy of the Mundane: Why the Muggles Prefer Not to Know. Professor Egg argues cogently that Muggles are much more likely to believe they have misheard something than that they are hallucinating.
So the Wizarding Wireless Network runs with magic, not electricity.
This is further shown when Mrs Weasley controls her wireless by magic rather than by using an electric remote control or something of that ilk.
They were all supposed to be listening to a Christmas broadcast by Mrs Weasley's favourite singer, Celestina Warbeck, whose voice was warbling out of the large wooden wireless. Fleur, who seemed to find Celestina very dull, was talking so loudly in the corner that a scowling Mrs Weasley kept pointing her wand at the volume control, so that Celestina grew louder and louder.
(Half-Blood Prince, Chapter 16, A Very Frosty Christmas).
Nevertheless, the restriction on using electricity around magic doesn't apply to all magical locations. It's mentioned solely in regard to Hogwarts. Presumably Hogwarts has more "magic in the air" than most other places because of the sheer number of people doing magic there.
"All those substitutes for magic Muggles use - electricity, and computers and radar, and all those things - they all go haywire around Hogwarts, there's too much magic in the air."
(Goblet of Fire, Chapter 28, The Madness of Mr Crouch).
It may well be that domestic homes like the Burrow aren't affected by this restriction.