Yes, much the same. See the full sheet music of the 1891 Adams version in The Lester S. Levy Collection of Sheet Music:
http://levysheetmusic.mse.jhu.edu/index.html. Hit Search and type in There is a tavern (or click here). The 1891 Adams version is more complete and closer to the one yowled by everyone over the last 75 years or more. The music is the same.
Of course we don't know which of the two named authors, or someone else for that matter, wrote the song. We have a publishing date for one version and an estimated date for the other, but this doesn't establish original authorship.
The traditional Ballad Index states there is no author cited on the 1883 printing.
The song booklet, with Tavern in the Town on p. 7 of Bodleian images of the 2-sheet booklet (no music), printed by Marsh ("between 1877 and 1884"), in London, has American songs about baseball teams, nigger songs, etc., i. e. songs cobbled from both sides of the Atlantic. An English provenance for the song cannot definitely be established. Music for the song by Day and Hunter, is mentioned along with W. H. Hills, but is this the same music we know?
I would like to see the Cornish "Alehouse" version, and will look for it, but there does not seem to be a copy in the Bodleian Library. Maybe Masato knows of it.