There's a recording on YouTube of TREAD ON THE TAIL OF ME COAT sung by Willie Brady from an album called "Irish Humorous Songs, Volume Two." No date is given, but the album cover looks old—I'm guessing 1970's based on the typography.

There's another version on YouTube called MUSH MUSH MUSH TURAL-I-ADDY, sung by Michael O'Duffy. The album is not identified, nor the year of release, but the smarmy orchestral arrangement makes it sound older.

The lyrics of both versions are closer to those posted by George Seto than to those sung by Moran & Andrews (and posted by me above).