Written by Tony the Troubador (aka Tony Stecheson) according to Ron Cahute. - Phone interview with Ron Cahute conducted by Danylo Centore 21 August, 2020. This turned out to be INCORRECT once the original Tony the Troubador 78 was tracked down.
I have so far been unable to find the original recording. However, there is a 1947 Toronto radio guide which mentions both Tony the Troubadour and the Anniversary Song, suggesting Tony was actively singing and aware of the song at the time it was popular.
Found the original record by the "Troubador" label cat# 2056. The credits on this label say "Ukrainian lyric - S. Mozewsky" - it is unclear who this refers to. Discogs: https://www.discogs.com/Tony-The-Troubador-Anniversary-SongThe-Cossack-Rides-To-Battle/release/8927477