I am sure that the threads containing children's songs at this website contain some documented older songs.
Threads 585, 2910, 4300, 8650, 18352, 20909, 35097, 37840, 53849, 56303, 56361, 56380, 60248, 61925 and others.

The Cariboo Gold Rush brought people from all over, but an Ontario girl would be most familiar with UK-Ireland and eastern U. S. songs. (Johnny Appleseed was a Johnny come-lately as noted above and only crabapples can be grown in much of the western prairies that she would cross).
Versions of "Lord Randall" (thread 10062), "Poor Caroline of Edinburgh Town," "Drowsy Sleeper" and other old ballads, some of the songs of Robert Burns,- quite a lot available. At age thirteen she would have heard some of these ballads and songs, and become interested in murders and ghosts.
Undoubtedly known would be "Sweet Betsy from Pike." "Down by the River " (Clementine) didn't show up until 1863. "She'll be Coming 'round the Mountain" had its roots in a Negro spiritual, but was spread everywhere. "Froggy Went A-Courtin'," "Courtin' in the Kitchen," "Shule Agrah," "The Jolly Beggar."
As mentioned before, many of the Foster songs were popular in eastern Canada as well as in the United States and British Isles.
Undoubtedly sung, but not sung today, would have been blackface minstrel songs (there are pc versions of Kemo Kimo) which were widespread in both the British Isles and the U. S.