@ChillDoubt
Yep, *that* one. You Da Man!
I posted the OP while the prgram was still being transmitted. I couldn't link to iPlayer and some programmes take their sweet time about materialising. I had to write the bit about "40 minutes in" before the day was out or I would have forgotten by the morning.
I apologise to all and sundry for my impatience in setting the question and being so narrow in what was acceptable as an answer but my threads routinely get shunned on this site and, once knocked off the unanswered tab, it would have drifted into obscurity. Like I said, this track has been popping up on TV for years and it drives me bonkers the way the beeb continually refuses to list simple minutes:seconds artist/title. Their producers love their favourite music, use it everywhere but won't credit it so we can find it and buy it.
Now I can chill out, thanks to ChillDoubt. How cool is that?