Disclaimer: I'm not familiar with either "Mary Ambree" or "The Blind Beggar of Bethnal Breen".

Robert Dodsley (1703-1764) wrote "a dramatic tale" titled "The Blind Beggar of Bethnal Green", either as a "dramatic performance" (1741?) or from which such a work was later prepared. The music was written by the famous composer Thomas Arne. The National Library of Australia holds a number of copies, some in online versions, but you must have an account through a "non-profit subscribing institution" (which I do not). I don't know whether some of them include Arne's music; some entries state explicitly that they do not. They also have "The History of the blind beggar of Bethnal Green", from an American source.

The Dodsley/Arne production was not a success, though some music from it achieved popularity.

Here's an entry from A Dictionary of the Drama.... (p.173) by William Davenport Adams, describing an even earlier work:
"Blind Beggar of Bethnal Green The), "with the Merry Humour of Tom Stroud, the Norfolk Yeoman." A comedy by John Day and Henry Chettle (q.v.), " divers times publicly acted by the Prince's servants " in 1600, and printed in 1659. In this piece the writers do not follow the well-known ballad. A second and a third "part" of "The Blind Beggar,' carrying on the story of Tom Stroud, were written by John Day and W. Houghton in 1601 (see Fleay). (2) A ballad farce by ROBERT Dodsley (q.v.), music by Dr. Arne, first performed at Drury Lane on April 3,1741, with Berry as the Beggar and Mrs. Clive as Bessy, his daughter. The Beggar wishes his daughter to marry Sir William Motley ; but her heart has been given to Wetford, who has rescued her from seduction, and the Beggar and Sir William acquiesce in her choice. The former reveals himself as Sir Simon Montford, and presents his daughter and her lover with £6000, which he has saved from the wreck of his fortune. Among the characters is John Sly, a Puritan. (3) A play in two acts, by H. M. Milner (q.v.). See Beggar Of Bethnal Green."

And from page 134:
"Beggar's Daughter of Bethnal Qreen (The). A comedy in three (?) acta, by J. Sheridan Knowles (q.v.), first performed at Drury Lane on November 22.1828, with Aitken as the beggar (Albert), Miss E. Tree as Bess (his daughter), Mrs. Knight as Elizabeth (his wife), Cooper as Lord Wilford, Vining as Lord Willoughby, W. Farren as old Small, Harley as young Small, Liston as Peter.yiTS. Faucit as Queen Elizabeth, Mrs.C. Jones as the Hostess, and Mrs. Orger as Kate. The prologue was written by Charles Lamb. The play was afterwards altered, rechristened'The Beggar of Bethnal Greenland performed at the Victoria Theatre, London, in 1834, with the author as Lord WUford, Miss Jarman as Bess, Miss P. Horton as Kate, Mrs. Egerton as the Queen, Abbott as young Small, and Chippendale as Strap. It was first produced at New York in December of the same year, with the author in the above-named part. See BLIND BEGGAR OF Bethnal Green."

A number of songs have been set to the tune (as the search results at the National Library of Australia reflect), and it's possible that one of these has been published in a collection with the tune. A large music library (such as at a university) may have scores of Arne's works, or provide access to Early English Books Online (based in Ann Arbor?).

A rather long ballad about Mary Ambree was published by bishop Thomas Percy in his Reliques of Ancient English Poetry. There is a black-letter ballad in the Pepys collection. She appears to be an obscure or mythical historical figure whose fame was perpetuated largely by the writers Ben Johnson and Samuel Butler.

Wikipedia has an article about Mary Ambree and the siege of Ghent (Gaunt) in 1584.

More to your purpose, read this blurb on the song (and particularly Janet Russell's version, which appears to be her own copyrighted creation) from Sing Out!, Summer 2009.