Actually an air composed by Thomas Connellan which fits very nicely as an intro to a fast reel (in F or Dmin). Played by Grainne Hambly (also in duo with William Jackson) where she combines this song with The Rectory Reel (Patrick Davey).
If you play this tune on a C whistle with the E minor setting, then you will be playing in D minor.
I am not familiar with your traditional music (Hungarian music is my hobby) though I like it.
I just tried an example of music search (also my hobby) as a potential contribution to this question.
Using connellan as text search in the Folktune finder http://www.folktunefinder.com/ I found five similar melodies, completely different from the two sheets, 4 with the same title and one with the additional word "air" (melody search with the notes of the sheet showed no useful result)
In Musipedia http://www.musipedia.org/ I found no usefull results in tekst search nor in melody search
Using connellan as text search in the Melodycatcher http://www.melodycatcher.com/ I found 7 results.
The first result is the same melody (in the same key as the first sheet). The six other results are the same melody as found in the Folk Tune finder part with the same title, part with the addition Air and Little respectively. For further details try the same tekst search
Using melody search 12 notes of one of the sheets were needed to find just that first melody
I have no Idea to what extend such results are relevant for the answer to the question, but if there are any further questions, please let me know.