This song just popped into my head, and although the melody is not the same as the mystery song in this thread the chord progression seems similar, and this page states that Ewan MacColl wrote it:

School days over

I might be barking up the wrong tree, but is it possible that Mr MacColl rewrote the lyrics to fit the topic of the documentary? It might also explain why it is so difficult to track down. It could be a one-off, project-related song.

(And just as an aside, I did an internet search for the phrase "down to Nottingham good prospects are my goal" and found the same request for info on Reddit, also posted by our GUEST,Brian_M, and the two AI/ChatBots said they could find nothing on the song and that ended the discussion, so we humans can pat ourselves on the back for caring enough to search for the answer to the mystery, even if we don't solve it. :-D )