The credits say that all the music is specially composed for the series and there's a list of composers, though no indication of who was responsible for that particular song.
If memory serves, it's Cara Dillon 'There Were Roses'. That's ceratinly on one of the series that the Big Yin did, but whether that was Britain or Ireland I can't remember.
Ralph McTell does the odd song too; he's an old friend of Billy's from years ago back when he was a folk singer who told jokes!
It was a lovely, lilting melody. I noticed in the end credits that "Mary Hopkins" was credited. Since she was/is a folk singer, it may well be one of hers.
Ah, got it! After a bit of searching, it's "England" written by Ralph McTell - there are various different recordings with altered lyrics and different singers used on the programme, but that's the song you're looking for.