From the notes to the Penguin Book (1959):

"This ancient ballad was a great favourite with singers in England and Scotland.  Sharp alone collected at least twelve distinct versions.  It was often printed on broadsheets and in America a good version found its way into a popular song book, The Pearl Songster¹, about the middle of the 19th. century.  Some texts make it clear that the bold girl had bewitched the lover into his deep sleep.  In England, other versions of the song have been reported from Dorset (FSJ vol.III, p.69), vol.IV, p.115, vol.VII, p.31), Lincolnshire (FSJ vol.IV, p.110), Somerset (FSJ vol.IV, p.112, vol. VII, p.33), Norfolk, Hereford (both FSJ vol.IV, p.114) and Essex (FSJ vol. VIII, p.127)."  -R.V.W./A.L.L.

This version was collected by Ella M. Leather and Ralph Vaughan Williams from Mrs. Powell of Weobley in Herefordshire, in 1910, and was first published in the Folk Song Journal, vol.IV, p.114.

¹Published by C. P. Huestis, NY, 1846.

Other versions on the DT:

The Broomfield Wager   Transcribed from a 1977 recording by Roberts and Barrand.  The version they sing came from source-singer Cyril Poacher of Blaxhall in Suffolk; he was a regular at the famous Ship Inn.  On Cyril's first recording of the song, made by the BBC (Child Ballads Vol I, Topic 12T160, 1961 [originally Caedmon]), singer and audience constantly interject the phrase hold the wheel.  This allegedly arose as a result of the singer trying to explain the story to a visiting yachtsman who misunderstood had his way as hold the wheel, but by the 1970s Cyril had gone back to the old way of singing it.  (ref.  notes to Cyril Poacher: Plenty of Thyme, Musical Traditions MT CD 303 ).
Two tunes are given; the second is Cyril Poacher's, the first (BROMFLD1) names no source, but is in fact the Penguin version.

The Broomfield Wager(2)  from Sam Henry, Songs of the People: with tune.

The Broomfield Wager(3)  Child's version C, originally published in Buchan's Ballads of the North of Scotland.  No tune.

In the Forum:

The Broomfield Wager  Some discussion, and a link to a transcription of the broadside version printed by J. Pitts of London.

There is an entry at  The Traditional Ballad Index:
The Broomfield Hill

Child #43
@witch @love @talkbird @magic

There is a version, no source given but apparantly collected in the Upper Thames Valley, at Ron Clarke's website:  The Broomfield Wager  A tune is also given; again, no source is named; just the remark that it "comes from Southern England".

I'm not as a rule going out of my way in these listings to link to transcriptions elsewhere on the web made from recordings by Revival singers, but will just mention that the text of Martin Carthy's 1965 version is available at:
Henry's Songbook  with notes added by Susanne, and
Gary Gillard's website

There are a number of broadside versions at  Bodleian Library Broadside Ballads.  Where there are duplications, I have given a link to the most legible copy:

The Merry Broomfield: or, The West Country Wager     Printed between 1711 and 1732 for Tho. Norris, at the Looking-Glass on London Bridge. License note: Licens'd according to Order.
The Merry Broomfield: or, The West Country Wager     Printer and date unknown
The Merry Broomfield: or, The West Country Wager     Printed and Sold by J. Pitts, No. 14, Great St. Andrew Street, Seven Dials, London, between 1802 and 1819.
The Merry Broomfield: or, The West Country Wager     The Merry Broomfield: or, The West Country Wager  Printed and Sold by John White, in Pilgrim-street, Newcastle, between 1711 and 1769.

The Merry Broom Fields: or, The West Country Wager  Printed by H. Such, Printer & Publisher, 177, Union Street, Boro' S.E. And 5, Back Church Lane, Cable-St., St. George's East, London, between 1863 and 1885.
The Merry Broom Fields: or, The West Country Wager  Printed by J. Turner, High Street, Coventry, between 1797 and 1846.

These are all large images.

Malcolm