Having listened closely to all the Radio Ballads (The new ones), I'm sure that the technique used was to record the songs in one go, but with carefully timed (to the second probably!) instrumental interludes.
All the music would then have been mixed together, thus making a music "bed" over which the speech inserts (already selected and pre timed) could be overlaid. A relatively simple process using digital editors. When I was working on Radio 1/2 docs in the 70s and 80s...It was loads of reels of tape, that you normally kept losing!

If you just released the basic music "backing" as used on the programme it would sound quite odd I'm sure!!!
Mind you, as you say there are some fine songs there, and I'm sure that more conventionally structured versions will find there way onto the artists next projects in due course.