On www.nightclub.com, an entertainment website that features articles on drinks of various kinds, it is stated (back in the November 1998 issue) that Jim Beam's rye was first released in 1940, and Wild Turkey rye, in the late 1940's.
I am sure that the producers of Jim Beam and Wild Turkey bourbon (or their antecedents) also produced rye before 1919 but as reconstituted after 1933 these operations waited until 1940 and the late 1940's to put out these two brands.
Any theories why? Maybe they felt they had to concentrate on bourbon first (from the early 30's) and once they saw its foundations were established they could turn to rye. This decision must have been made from about 1935 for Beam's rye (to have four year old whiskey to sell in 1940) unless possibly Beam purchased bulk rye and sold that under its name to see if brand acceptance could be secured before manufacturing the rye itself. (Certainly even in later years some large distillers, e.g. National Distillers before the sale to Jim Beam, sourced their rye from various sources, e.g. Pennco (later Michter's) in Pennsylvania. Since rye was typed as a Northeast/Mid-Atlantic Seaboard drink maybe the release of Kentucky rye in the 1940's showed the growing strength and ambitions of the larger Kentucky distillers..