OK, just 21 years late... I have "The Whole World Round" on the "Woodstock Mountains" LP sung (beautifully) by Lee Berg and Rory Block as noted by the first poster, where it is credited as traditional; and also on the Dillards' "Live-Almost" (1964) where it is credited to "J. Stuart, M. Jayne", Mitch Jayne being the Dillards' bass player and front man of the day. The Stuart/Jayne authorship is confirmed by Rodney Dillard in this interview where he says: " "The Whole World Round": What happens when you can hear your neighbor’s car start up or the low end wolfer on somebody’s jeep who’s two miles away? A lonesome song, written by Mitch and Joe Stuart in 1964..."

Mitch Jayne, together with Dillards' mandolinist Dean Webb, also penned "The Old Home Place", another (possibly more frequently heard/recorded bluegrass original now approaching "standard" status, I would say...) ... I heard both of these first in the early 70s, "The Whole World Round" on my copy of the Dillards album, and "The Old Home Place" (which appeared first on their previous album, "Back Porch Bluegrass" from 1963) floating around in sessions.

In another portion of his Jeremy Roberts interview (initial portion cited above), Rodney says:

"Jayne was also a talented songwriter and novelist who rescued a number of the Dillards’ earliest shows in front of indifferent audiences utilizing sharp comic timing. Jayne co-wrote the majority of the band’s essential songs — “Dooley”, “Old Home Place,” “There Is a Time,” “The Whole World Round,” “Nobody Knows,” “The Biggest Whatever, “Listen to the Sound,” “Ebo Walker” — and remained a staunch outdoorsman until his death in 2010."

Just in case anyone is interested!