In 1935, Harry Bernstein published the idea of switching a bill with a bill-sized piece of paper, and then back again. See “Paper-Mint” in The Sphinx, Vol. 33 No. 11, Jan. 1935, p. 335. His method involved folding the paper into a small packet and exchanging it with one already folded inside a thumb tip. Bernstein left out many handling details, so it's difficult to gauge how close his technique was to modern approaches, but the general concept was certainly in play. He claimed to have based it on methods used to restore torn pieces of paper. The obvious precursor would be Ching Ling Foo's torn and restored paper ribbon effect, which was highly popular during the late 1800s and first decades of the 1900s. Ching was said to have used different methods, one of which involved a reversible thumb tip of cloth for the switch of papers. Al Meiners is said to have first employed the metal thumbtip to accomplish Foo's effect (see E. G. Ervin's Club Deceptions, 1947, p. 47), although Meiners was mistaken in believing he invented the gimmick. Prof. Herwin preceded him with its invention c. 1885.
It appears to have taken four decades after the publication of Bernstein's “Paper-Mint” for the bill-transformation trick to catch on. The resurgence was sparked by Eastern-European circus performer Vladimir Vladimirov, who developed the now-common handling for the thumb-tip switch of bills. The idea caught on with magicians when, in 1976, his technique was taught to Mike Kozlowski, who made some alterations and published his handling as a monograph titled The Hundred Dollar Bill Switch, 1977.
A full history of the bill switch, along with an encyclopedic look at the technical and presentational variants was published in John Lovick's Switch, 2006.