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). The original German lyrics were by Florenz Friedrich Sigismund with melody by Friedrich-Wilhelm Möller.
By 1966 a song titled "Я є веселий мандрівник" existed in the USA (source: Юнак 1966-11 (ст 24)). This version was clearly a translation of the English translation "I am a happy wanderer" by Antonia Ridge. It is unclear what lyrics were actually being sung at that time - the earliest set of lyrics under the "Я є веселий мандрівник" 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 "Я є веселий мандрівник" 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: This song 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.