The earliest known version of the lyrics were written by Ostap Hawaleshka in 1960 with the title "Мій Батько був мандрівником" as a loose translation of the German song "Mein vater war ein wandersmann / Der fröhliche wanderer" ( https://www.youtube.com/watch?v=hx0r0R6q5YQ ) which he had heard on the radio in Canada. The song became massively popular and by 1961 Plastuny in Buenos Aires were already singing it. (Source: Phone call between Danylo Centore and Ostap Hawaleshka Dec 6, 2020)
By 1966 a song titled "Я є веселий мандрівник" existed in the USA (source: Юнак 1966-11 (ст 24)). It is unclear what lyrics were actually being sung at that time - the earliest set of lyrics under this title that I've found are from 1996. Today, "Я є веселий мандрівник" is the more popular variant and while the lyrics are substantially different than Ostap's, there are a few lines which show some influence, e.g.:
Ostap's lyrics: І так я буду мандрувать аж поки я не вмру Modern lyrics: Я доти буду мандрувать, аж доки не умру
I do not currently know who was responsible for the bulk of the modern lyrics. Perhaps there were actually two independent translations which gradually merged or someone couldn't remember most of Ostap's lyrics and filled in the blanks.
Fun fact: It was sung in 1989 in Ukraine (Svoboda 1989-172 (ст 2)) with mixed Ukrainian and English verses. The lyrics of this variant are currently unknown.