While they were manning the capstan in the movie, they were singing "Yo ho! Yo Ho! Yo ho!" etc. followed by inaudible words. It became slightly more tuneful and may have turned into "Yeo, Heave Ho!" but I wouldn't bet on it. It didn't sound like "Fifteen Men" to me.

The lack of the "Hurrah!" in the second chorus of "Goodbye, Fare You Well" was surprising.

At any rate, as I hinted and MtheGM observes, "modern" shanties are anachronistic in the "Treasure Island" era.

And they weren't sung for diversion to the accompaniment of a lutelike instrument.

CB, my notes tell me that Terence Stamp (or a voiceover) actually led the singing of "Hanging Johnny" in Billy Budd (1962) one of the finest Hollywood screen adaptations ever. He could have sung the equally anachronistic "Reuben Ranzo," but that wouldn't have foreshadowed his own hanging.

"HJ" is also audible in the background of Mutiny on the Bounty (1935).

(I knew those notes would come in handy!)