My copy of the Roud index lists 35 entries for the song (#446), all from the USA, the earliest versions seem to be collected by Sharp in 1917 and 1918 (both starting My father always told me..). (These are both given as MSS Folk Tunes, so I don't know if that means there is no complete text with them).

The titles are varied: many different spelling of Rose Connally and similar, Willow Garden or Down in/by the Willow Garden being the most common.

Mick