Probably published in the 1950s or even the late ’40s. Maybe in Britain or Canada.
Children’s fantasy/detective story
Plot: A single (gasp!) mother (probably a war widow) with 2 kids, a boy and a girl. She is being courted by the neighbourhood beat cop. The kids want to buy her something special for her birthday and they have saved their allowance so they have enough to buy her a small second-hand radio. The kids know she loves music and dancing. Kids find out later that radio is magic and broadcasts crimes in progress, which they , of course, try to stop/solve, and get their bacon saved by the cop character. I know this is a faint, faint hope. But it was such a change from Dick and Jane, which I refused to read aloud, thus grade one ended with me being labelled as mentally deficient. It was astoundingly different from anything else I ever saw until I was in my teens.Best of Luck to you – and for me!
‘The Radio Imp’ by Archie Binns.
A 1950 review of the book is as follows: ‘The goal of Jim, Mary and Margaret is a radio. When the goal is finally reached the result is a character reformation in cranky Mrs. Flanagan, a romance between the children’s mother and Mr. O’Donnel, a member of New York’s Finest, and the capture of a criminal for whom a sizeable reward is offered. A delightful family spirit of sharing and solidarity pervades the book. Chores, responsibility and money are shared. The children all conspire to protect their adored mother, who works as a millinery saleswoman.’
There are some copies available, starting around 50 US dollars. Try ebay, etsy, bookfinder(dot)com