There are (at least) two "Bold Fisherman" songs, one a folk song (notably sung by the Coppers and Walter Pardon) and one a music hall song which was written by George W. Hunt in the 1870's or earlier; see this thread for more info, including (close to) the original text and tune. Bogie sang a version of the Hunt song, though the latter two verses and the latter part of the tune seem to be Hollywood invention, twisted well beyond what usually happens in folk "evolution". Whether Bogie actually learned it from his grandmother (as he said when introducing the song later on a radio broadcast) is dubious--I believe the song was cockeye-cocted into the script, and Bogie learned it from there.

So, if it was a clip of Bogie, or of someone singing the same verses Bogie sang, there probably was copyright infringement; regardless of whether anyone profited from it monetarily, damage was ostensibly done by making the song widely available gratis, without permission and without recompense to the arrangers. If it was Hunt's version or the folk song, you're safe (barring transgressing someone's copyrighted arrangement). If it was some other version of the Hunt song and your version is closer to Bogie's tune than Hunt's tune, you may have to do some research to prove there was no infringement.

Disclaimer: I'm not a lawyer, and this advice is only worth the paper it's not printed on.