Not sure if this counts, because June Tabor didn't write it (Les Barker did) and Les Barker doesn't sing it (June Tabor does), but "The January June" is about the one who sings it and also mentions the one who wrote it:

The January June she walks abroad
in sombre mood and boots of leather,
The February June she sings a song
About a pigeon getting lost.
The June of March may briefly smile,
But only for a little while;
It's good to get it over.


In April, it's the Easter Tree;
This June's degree's in misery; a master's;
In May the June of May may say
The June of June is due this afternoon.
Too soon the June of June's come true
To do as June Junes used to do;
Les Barker, you're a bastard.

(There's more, elsewhere in the forum)