Starting as early as 1898, at least, THE GOLDEN CITY was published in hymnals. Variants of its text go back to some time before 1882. Its tune is an adaptation of the chorus of Peacock's WE WILL WALK IN THE STREETS OF THE CITY (1874). THE GOLDEN CITY has "Words arr. by Geo. J. French" and "Geo. J. French, Arr." (of the music). It seems evident that French based his tune "arrangement" on Peacock's chorus. It also seems likely that the text of GOLDEN was inspired by Flamman's text for STREETS, the GOLDEN text probably being a rewrite aimed at Adventists.
French removes the "jerkiness" from Peacock's tune by replacing dottted-eighth-sixteenth note pairs by eighth-eighth pairs. He also modifies the incipit, from SOL-do-mi (STREETS and RED RIVER VALLEY) to do-re-mi (GOLDEN). My recollection is that New Orleans jazz bands follow GOLDEN and play do-re-mi, rather than SOL-do-mi.
My guess, then, is that jazz musicians picked up the song from French's arrangment of GOLDEN. The latter was arranged from the chorus of Peacock's STREETS. GOLDEN uses variants of the same tune for stanzas and chorus, as do, I think, jazz bands. STREETS, however, has different tunes for stanzas and chorus.
RIVER seems to have taken its tune from the chorus of STREETS, while jazz bands take theirs from GOLDEN.
Interestingly, Taylor's 1882 song, also entitled THE GOLDEN CITY, follows Peacock in having different tunes for stanzas and chorus. Moreover, Taylor's tune for the chorus seems to contain a feature of the usual RIVER tune that both Peacock and French lack. P and F have the last two notes of the 3rd phrase as fa-fa, whereas RIVER and Taylor have the as sol-fa. ("Just remember the Red River Valley" - these are the notes for "Val-ley".)