This seems to be the same song as 'Go From My Window', popular and widespread in the 16th and early 17th centuries. The earliest known text is in John Fletcher and Francis Beaumont's play of 1611, 'The Knight of the Burning Pestle' (Act III, scene v). Because the song was already well-known and thus served a purpose in the play - a common ploy in Shakespeare's time - the text is partial, giving only the first verse. By this time the tune had been used frequently by composers as the basis for sets of variations (see below).

The (tune and) text used below predates the melody and words used by folk singers today.

1. Go from my window, love, go
Go from my window, my dear
The wind and rain
Will drive you back again
You cannot be lodged here

2. Go from my window, love, go
Go from my window, my dear
The wind is in the west
And the cuckoo's in the nest
You cannot be lodged here

3. Go from my window, love, go
Go from my window, my dear
The devil's in the man
And he cannot understand
That he cannot be lodged here


The more usual rendering today is more like this:

1. Go from my window, my love, my dove,
Go from my window, my dear,
For the wind is in the west and the cuckoo's in his nest,
And you can not have a lodging here.

2. Go from my window, my love, my dove,
Go from my window, my dear,
O the weather it is warm, it will never do thee harm,
And you can not have a lodging here.

3. Go from my window, my love, my dove,
Go from my window, my dear,
The wind is blowing high, and the ship is lying by,
And you can not have a harbouring here.        

4. Go from my window, my love, my dove,
Go from my window, my dear,
The wind and the rain have brought him back again,
But he can not have a harbouring here.

5. Go from my window, my love, my dove,
Go from my window, my dear,
The devil's in the man, that he will not understand
That he can not have a lodging here.                 

This is one of a very small number of traditional songs that have genuinely survived in tradition continuously since the Renaissance. (The only other example I can think of is The Three Ravens that appeared in Thomas Ravenscroft's Melismata, 1611.) The song must already have been well-known when, in 1567, it was used as a template for an anti-Catholic song in The Gude and Godlie Ballatis, collated by Scottish editors. A version of the ballad, Goe from the windowe goe, was licensed in England on (the later than Gude and Godlie date of) 4th March 1588, but it is lost unless parts of it are echoed or reproduced in Merrythought's song in Francis Beaumont and John Fletcher's play of 1613, 'The Knight of the Burning Pestle', where the first set of surviving words appear.

There is a mass of evidence for the huge and continued popularity of the song. Renaissance lutenists and harpsichordists continually set variations on the tune. To name but a few: anonymous in the Folger Dowland MS, c. 1590; John Dowland in William Barley, A New Booke of Tabliture (for orpharion), 1596; Thomas Morley (1557-?1603) in his First Book of Consort Lessons; John Munday in the Fitzwilliam Virginal Book, compiled by Francis Tregian in the early 17th century; Thomas Robinson in his The Schoole of Musicke, 1603; John Dowland again in the Jane Pickeringe MS, 1616. There was also William Byrd, Orlando Gibbons, Edward Collard ...

Still later, and now in the Baroque period, the playwright John Fletcher refers to it in two other plays (besides Burning Pestle): Monsieur Thomas (1639) and The Woman's Prize (1640). Later still, a rewritten text appeared in Thomas D'Urfey's Pills to Purge Melancholy, set to the tune of Tomorrow is Saint Valentine's Day.

Due to its widespread popularity amongst Renaissance virtuosos, we have no doubts as to the original tune. The melody that has survived to this day in the oral tradition is remarkably similar to that used by Elizabethan composers, and the first known words have also changed little.

In oral tradition, this song has been collected from James Parsons in the 19th century by Revd. Sabine Baring-Gould; and from persons unknown (to me) by William Alexander Barrett, and by Peter Buchan (published 1828).

Hope this helps and is something like what you're looking for.