The "cherry" song is in the DT under the title "The Riddle Song."
I have received my first volume (Yahoo!) of the new edition of Child, The English and Scottish Popular Ballads (well-printed and with corrections). Child No. 46, Captain Wedderburn's Courtship, under C, Sheldon's Minstrelsy of the English Border, p. 555, includes the verse (6):
"You must get me to my supper a chicken without a bone;
You must get me to my supper a cherry without a stone;
You must get me to my supper a bird without a ga,
Before I go to Woodland's house and be a lady of your ha'."
The usual answers are in the following verse, and then she riddles him back.
The song, "Across the Blue Mountains," seems to be a small fragment of the original, with the riddles removed.