Irish Sergeant, one Belfast printing known. The Bodleian Library has no Green grow(s) laurels or lilacs. A Belfast printing of green grows the rushes 1846-1853. English printings back to 1813-1838.
Very little in Library Congress or Smithsonian with any age to it. A NY printing of Green Grow the Rushes (Scottish dialect in part) from 1860. Historic sheet music has nothing 1850-1920. Green grows the Laurel (lilacs, orange and blue, etc.) is listed at 1908 in the Traditional Ballad Index.
The scarcity of broadsides and sheet music suggests that this song, except in the Scottish version, was not all that widespread, and little known in America. All references suggesting age (other than those cited) are seemingly without support.
With a search, it may be found as a mention in diaries and letters of the time, or in Irish archives, but at present, its popularity during the Mexican, or even the Civil, War, is doubtful.