I am the Kat that Walks By Itself

I think that UK radio stations aimed at the general public, and broadcasting news and/or entertainment, began during the 1920s. For example 2LO, which became the British Broadcasting Company and subsequently the British Broadcasting Corporation (BBC) went on the air in 1922, and was the second British station. (Info from the Wikipedia article on 2LO.) So I'd suggest the mid 1920s being a plausible date for the beginning of the wizarding wireless.