Tommy Tucker published this plot – using a glide for the false count – in Chas Eastman's Expert Manipulative Magic, 1933, p. 18. (See pull-down reference for citations on the use of that sleight by John Booth as a subsequent sleight-of-hand method for this trick.)
The first published suggestion of using bills instead of cards seems to be by C. Brustia in The Sphinx, Vol. 36 No. 1, Mar. 1937, p. 8. Brustia featured only the idea of using bills instead of cards, while writing, “Use your favorite method.” Brustia preceded Tom Bowyer and his “Repeat Bill Trick” in print by nine months. Bowyer contributed his version to Genii, Vol. 2 No. 4, Dec. 1937, p. 110. Sid Lorraine credits Baffles Brush with the card-envelope method, in the late 1930s (see Tops, Vol. 18 No. 5, May 1953, p. 19). He writes, “The method, however, was first introduced to us years 'n years ago by the late Baffles Brush. It was during John Ramsay's first trip to America, which should date it. Baffles disclosed the method to John and yours truly in deep secrecy. Several inventors have claimed the method since.” Bowyer used envelope bills in his aforementioned “Repeat Bill Trick.” Since Bowyer was Canadian, he may have been in Lorraine's circle and privy to this information. This first trip by Ramsay to America seems to have been in 1934, when he performed at the IBM convention in Batavia (see The Linking Ring, Vol. 14 No. 4, June 1934, p. 275).
Sam Berland came up with the idea of using rubber cement on the edges of the cards to hold packets of three together, publishing it in The Tarbell Course in Magic, Vol. 6, 1954, p. 75.