My earliest musical memories were from the "wireless" (no records or TV in our house in those days). We were living in Glasgow and, thinking back, it was all a mixture of stuff from the BBC Scottish Home Service and the BBC Light Programme. So...

Harry Lauder singing "Keep Right On To The end Of The Road"
Kenneth Horne & Richard Murdoch singing "It's Much-Binding-In-The-Marsh" (theme to a comedy half-hour)
Theme music to "Listen With Mother", "Worker's Playtime", "Housewives Choice", "Five To Ten" - and stuff like that.

The first song I heard my mum sing was (to the tune of "There is a happy land"):

There is a happy land, far far away
Where little piggies run, three times a day.
You should see those piggies run, when they see the butcher come.
He cuts slices off their bum, three times a day.