1903        Stone, Herbert Lawrence. "The Reckoning: A Story of the Sea." Short Stories vol. 52 (Oct-Dec. 1903). Edited by Alfred Ludlow White. New York: The Current Literature Publishing Co. 190-

Though a fictional short story, the chanties mentioned would seem to be based in reality. The material looks original, at least.

The story concerns a ship bound out of Frisco.

[LEAVE HER JOHNNY] is set at the capstan.
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This Tam-o'-Shanter was anchored in the stream not far from the Vigilant, and as Captain Bradshaw was put aboard his own ship again, he could see her sixteen men gathered on the top-gallant forecastle, their bodies bent over the capstan bars as the cable was hove in. And the refrain of the chanty that arose therefrom and drifted across the narrow stretch of water to the listeners on the Vigilant, ran:
          —"Leave her, Johnny, leave her. 

Oh, there's six feet o' water in her lower hold, 

So leave her, Johnny, leave her."
//

Later capstan songs are "Down the Bay of Mexico", which likely refers to this song,
http://www.youtube.com/watch?v=OilQra0NlRg
and "Walk Her Round" and "West Australia" ([SOUTH AUSTRALIA], I suppose]. And at the halyards there is [JOHNNY BOWKER] (not a customary use?) and [TOMMY'S GONE].
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Soon the click of the iron pawl dropping into place drifts aft, then the words of "Down the Bay of Mexico" rise in loud, crude tones, followed by "Walk Her Round" and "West Australia," to the rhythm of which the shuffling feet keep time. The iron cable comes slowly in, a link at a time, grating harshly on the hawsepipe, the mate now leaning out on the bumpkin to watch it, now admonishing the men to "walk her round briskly." Suddenly he straightens up, raises a hand to the men to cease heaving and shouts aft: "Up and down, sir!"

"Break her out, Mr. Dunning," answers the captain, and the bodies bend lower over the bars and muscles swell as the strain on the capstan increases. The songs have ceased and in their places are heard, here and there, the muttered words "Heave and raise the dead," "Dig your nails in, now," "Break her out." Slowly the anchor leaves its bed at the bottom of the bay and when it is at last clear and the strain on the cable is eased, the men break into a run and soon have it, dripping and muddy, hanging at the fore-foot…

…The wind being fair, the gaskets are soon off the topsails and the sails sheeted home. The upper topsails are mastheaded to the tunes of "Johnny Bowker" and "My Tom's Gone to Hilo," the ex-boarding master being driven from one halyard to another, where he "tailed out" with the crew as well as his aching arm would allow.
//