I've made midis of the two sourced versions given in this thread:
(1) The Moon Shines Bright Noted by Lucy Broadwood from the Goby family of Sussex/Surrey.
(1) The Moon Shines Bright Noted by Cecil Sharp from Mrs. Prudence Handy of Ilmington, Warwickshire, in 1910. You'll notice that this tune is very close to the one recorded by Martin Carthy, as he remembered it from Cynthia Gooding's recording; allowing for normal changes consequent on that kind of transmission, this could very well be the source of both.
Both midis will go to the Mudcat Midi Pages: until they appear there, the midis may be heard via the South Riding Folk Network website.
Malcolm