Two editions of Sep. Winner's 1868 sheet music (with a different music) are at The Lester S. Levy Collection of Sheet Music:

(1) Title: The Amateur. A Collection of Popular Songs, Ballads, Etc. Ten Little Injuns. Comic Song and Chorus.
Composer, Lyricist, Arranger: Arranged for the Guitar by Sep. Winner.
Publication: Philadelphia: Lee & Walker, 722 Chestnut St., 1868.

(2) Title: Ten Little Injuns. Comic Song and Chorus.
Composer, Lyricist, Arranger: Arranged for the Guitar by Sep. Winner.
Publication: Philadelphia: Lee & Walker, 722 Chestnut St., 1868.

Also at Historic American Sheet Music:

(3) Ten Little Injuns (Boston, Massachusetts, Oliver Ditson, 1868)

(4) Ten Little Injuns (Boston, Massachusetts, Oliver Ditson, 1896; 1868)

(5) "Ten Little Injuns" by A. Schuman (Saint Louis: Balmer & Weber, 1873; without words) at American Memory has an almost familar melody.

British broadside editions are at Bodleian Library Broadside Ballads:

(6) ten little niggers [title] (written by Frank Green)

(7) ten little ministers [title] (parody)

Members of this tune family includes "The Monkey's Wedding" (Monkeys Wedding, New York: Firth & Hall, n.d.; in Sandburg, The American Songbag, p. 113), "Paw Paw Patch" (click here), and a version of "Drunken Sailor."

References: Opie, The Oxford Dictionary of Nursery Rhymes, 2nd ed., no. 376; Fuld, The Book of World-Famous Music, 5th ed., pp. 205-06.