So far, the oldest (Chrome, Sheffield, 1817; Birt, London, 1828, title "The Maiden's Complaint"; 1834, and possibly Catnach, date (?)) all refer to the boy as Jemmy; forming a related group.
More versions may be in the Bodleian or other repository under names such as "The Maiden's Complaint," which aren't recognizable as "The Drowsy Sleeper" unless they are looked at.

"Jemmy and Nancy of Yarmouth" tells the same story of separation and cruel parents, in an overlong ballad printed by Bebbington in 1858-1861, Bodleian 2806 c.16(319). Other Jemmy and Nancy ballads are unrelated.
Jemmy

Also see "The Cruel Father," undated, Harding B25(450), and especially "The Cruel Father, or the Maiden's Complaint," Harding B25(452), pub. 1819-1844, Pitts and Toy, which begins "Awake, awake, you drowsy sleeper," another one with Jemmy; a different printing of "The Maiden's Complaint" from the one already transcribed.

There may be a relationship with some versions of "The Cruel Father and Affectionate Lovers," where the lover is a servant or clerk; the locale may be Shannon or England (1813-1838); and The Cruel Father and the Constant Lover, 1802-1819, 2806 c18(76) and others. These involve a dungeon and probably are separate ballads.
Also see "The Cruel Father," printed Edinburgh, Firth c12(298), sems to be fairly late 19th C., "a young sailor".