The first recording of I Won't Care (A Hundred Years From Now) was made by husband-and-wife Country duo, Bob Atcher and Bonnie Blue Eyes (real name Loretta Applegate) on 18 June, 1940 (source: Country Music Records : A Discography, 1921-1942 by Tony Russell, Bob Pinson; p.69). The recording was released on at least three different labels at different times:
OKeh 05755, August 1940;
Conqueror 9595, January 1941;
Columbia 37751, August 1947.
Here’s a brief soundclip of Bob Atcher and Bonnie Blue Eyes’ original recording of I Won’t Care (A Hundred Years From Now) on the OKeh label. Note that Dave McEnery’s surname appears in the writing credit.
The full recording can be enjoyed at https://archive.org/details/78_i-wont-care-a-hundred-years-from-now_bob-atcher-bonnie-blue-eyes-mcenery_gbia0059231a