From what I can tell, Arthur Godfrey's recording career mostly dates from the 1940's, and the late 40's at that. He recorded "Trail of the Lonesome Pine" in 1947 (Columbia 38246). But the song itself was popular long before that. It was written in 1913, and a cylinder recording from the same year, of Manuel Romain, can be heard here:
http://cylinders.library.ucsb.edu/mp3s/0000/0717/cusb-cyl0717d.mp3
Here's a recording of Vaughan Hughes and Stuart Morton from a 78 pressed the same year:
http://www.youtube.com/watch?v=V2aFd5hlTJI

The song was inspired by a popular novel of the same title by John Jay (1908). The book gave rise to a play, which featured the song, and at least three movies of the same name (1916 [a Cecil B. DeMille silent], 1923 [another silent] and 1936). The 1936 film, with Fred MacMurray, Henry Fonda and Silvia Sidney, won the Venice Film Festival Award for Best Color Film--being the first film shot in Technicolor--and an Academy Award for Best Original Song (for "A Melody for the Sky"). The MacDonald/Carroll song backs the opening credits, so the film was undoubtedly instrumental in re-popularizing the song.

The song got another boost the following year (1937) with the release of Laurel & Hardy's film Way Out West, where the pair sing the song in a bar:
http://www.youtube.com/watch?v=PZyU7tHO_1E
The song, as originally written, and as performed by Romain, L&H and just about everyone else, is all about June, and says nothing about a cow--a version I've never heard! (Though I may have lived a sheltered existence.)

Way Out West also features one of the most famous L&H moments, a silly dance they do upon their arrival in the cowtown:
http://www.youtube.com/watch?v=_SvuNcPx2dg

So if Godfrey "popularized" the song, he was only riding some very long coattails--which you can see in the clips.