From the notes to the Penguin Book (1959):

"The story is a legendary re-working of historical fact. Jane Seymour, wife of Henry VIII, died on 24 October, 1537, twelve days after the natural birth of her son, who later became Edward VI. Some said her death was due to clumsy surgery. We do not know how old this ballad is, nor if it derives from a piece called The Lamentation of Queen Jane, licensed for publication in 1560. The ballad has been collected in Devon (FSJ vol.II [issue 9] p.222) and Somerset (FSJ vol.V [issue 20] pp.257-8), and a second Dorset version is given in FSJ vol.III [issue 11] p.67." -R.V.W./A.L.L.

This version was collected by H.E.D. Hammond from Mrs. Russell¹ of Upwey in Dorset, in 1907, and was first published in the Folk Song Journal, vol.III [issue 11] pp.67-8.

There is another version on the DT:

Queen Jane from The Peggy Seeger Songbook, with tune: no original source specified.

There is an entry at The Traditional Ballad Index:
The Death of Queen Jane

Child #170.
@birth @history @royalty @death

There is a version at Lesley Nelson's Child Ballads site:
The Death of Queen Jane

There is a PDF file of the version collected by Baring Gould from Sam Fone, 28th. March 1893, with staff notation, at Sabine Baring-Gould and the folk songs of South-West England:
The Death of Queen Jane

The text published by the Percy Society in 1846 is available at Poets Corner:
The Death of Queen Jane

There is a broadside version at Bodleian Library Broadside Ballads:

Death of the Royal Queen Jane Printed c.1820 by W. Collard, Bridewell-Lane, and Hotwells, Bristol. (Large image)

There is a brief biography of Queen Jane at http://www.tudorhistory.org :
Queen Jane
and some pictures here.

¹ Mrs. Russell was an important carrier of tradition: other songs of hers available on the DT are:

The Bedmaking
Catch-Me-If-You-Can
John White
One Night as I lay on my Bed
Ye Mar'ners All

Malcolm