The song is believed to be an early version of “All round my hat”. Although it does not mention a hat it has some other elements such as a garland of green willow and separated lovers.

You can read Othello – by William Shakespeare Act IV. Scene III at:

http://www.readprint.com/chapter-7922/William-Shakespeare

The following is an extract:

DESDEMONA
[Singing] The poor soul sat sighing by a sycamore tree,
Sing all a green willow:
Her hand on her bosom, her head on her knee,
Sing willow, willow, willow:
The fresh streams ran by her, and murmur’d her moans;
Sing willow, willow, willow;
Her salt tears fell from her, and soften’d the stones;
Lay by these:–

Singing

Sing willow, willow, willow;
Prithee, hie thee; he’ll come anon:–

Singing

Sing all a green willow must be my garland.
Let nobody blame him; his scorn I approve,-
Nay, that’s not next.–Hark! who is’t that knocks?

EMILIA
It’s the wind.

DESDEMONA
[Singing] I call’d my love false love; but what
said he then?
Sing willow, willow, willow:
If I court moe women, you’ll couch with moe men!
So, get thee gone; good night Ate eyes do itch;
Doth that bode weeping?

There is a version of “All Round my Hat” at:

http://www.folkinfo.org/songs/displaysong.php?songid=151

Thx~
Ed