A.L.Lloyd (Folk Song in England, 113)quotes the fragment (which is what it is--Chappell gives only this, from the early 1500s song-book) and adds "This is always printed as a stray verse from a lost poem. But among H. E. D. Hammond's manuscripts is a night-visit song collected at Puddletown, Dorset, in 1905, which may represent the complete form of the piece. It is a version of the familiar ballad of the cock that crew too soon and made the lover turn out of his sweetheart's warm bed into the cold windy night." He gives the identifying verse, which is no. 6 in Hammond's MS. -- and that is available in James Reeves, The Everlasting Circle (1960), p. 136, as follows:

Once I loved a lass and she loved not me
Because I was grown poor, poor a little, poor a little, poor,
I knocked both loud and sure, sure a little, sure a little, sure,
My love she arose and slipped on her clothes
And came down and let me in.

When I beheld my true love's arms
My heart grew cold and faint, faint a little, faint a little, faint.
I took her round her middle so small
And carried her to bed.

All the fore-part of the night
We did both sport and play, play so pretty, play so pretty, play,
And all the last part of the night
She sleeped in my arms till day.

My love she kept a cock and a pretty crowing cock
And it crowed in the morning so soon, soon so very, soon so very, soon.
My love she thought 'twas day and she hastened me away
But it proved to be the light of the moon.

The wind it did blow and the cocks they did crow
As I tripped over the plain, plain so very, plain so very, plain.
So I wished myself back in my true love's arms
And she in her bed again.

Now I'll prove so true to my own true love
As the stars all in the sky,
And if she should not prove the same by me
She's far better lost than won.

The 4th verse is taken from another version, since HEDH just wrote "etc." after line 1. Bowdlerising, I suppose.

This strikes me as being very likely a descendant, at least, of the song of old Henry;s time.