I remember some of these from the mid 50's at Woodhatch Junior School near Reigate. I had forgotten the BBC programme but the thread has kick-started the brain! Now retired, I play guitar and sing around care homes on the Isle of Wight ( where I now live) and I already sing some of these songs to the residents. I plan to include a few more! Sadly, quite a few of the old people suffer from the various forms of dementia, but it's fascinating to see how these old tunes and songs stimulate them to take part.
Music and Movement? I have slightly different memories of that; I'm not sure if it would be politically correct these days to have the boys dancing around in their school shorts but the girls having to tuck their skirts into their knickers......