When I first heard this song in the 60s, it was introduced to me as a translation by A L Lloyd of a song collected by Bela Bartok.
Bartok and his contemporary Zoltan Kodaly were both active collectors of folksongs in the early 20th century in what was then Hungary. Today the boundaries have been redrawn to the extent that Bartok's birthplace Nagyszentmiklos in Hungary is now Sennicolau Mare in Romania.
Somewhere in my collection of tapes I have a recording of some of Bartok's collection sung in the original Hungarian, from a BBC radio broadcast many years ago. I think that Anna Thea (as I have always called it) may well be among them. The trouble is that I have never got round to cataloguing my tapes!
Anyway, I hope this contributes something, if only another line of enquiry.