As Louie Roy says above, the version of "Show Me The Way To Go Home" (or at least the lyrics) appears to have been written by Irving King (who also wrote "The More We Are Together") in or before 1925, when it was published. From Wikipedia ...

Irving King was a pseudonym used in the 1920s and 1930s by the British songwriting team of James Campbell and Reginald Connelly. Primarily lyricists, they generally worked in collaboration with composers.

It appears that they wrote both lyrics and melody in this case, though ...

The song was written on a train journey from London where both James Campbell and Reginald Connelly were travelling. They were tired from the trip and had a few drinks during the journey, hence the lyrics.

It appears that the song was pretty popular straightaway. It was recorded by Ella Sheilds(1879-1952) and by Wendell Hall(1886-1969), for the Brunswick label, in 1925, the year of its publication.