The chorus Hugill gives for the Limerick Shanty derives from Van Ambergh's Menagerie: A Comic Song, written for voice and piano by W.J. Wetmore and published in Brooklyn, 1865 (link). Isaac Van Ambergh (1811–65) was a lion tamer and circus pioneer famed for his travelling menagerie.

The song's chorus runs:

The elephant now moves round, the music begins to play,
Them boys around the monkey cage had better keep away.

The verses are amusing, but not limericks. I will leave it to another to compare the melody to Hugill.

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The song seems to have achieved immediate popularity. It is, for instance, one of the few songs without an explicit connection to the university published in Carmina Yalensia: A Complete and Accurate Collection of Yale College Songs (1867), with a slightly different text.

The song retained enough currency in later decades that it was parodied, for instance, in this February 1887 advertisement in the Cambridge Herald, an Ohio newspaper:

The Elephant now moves round,
The Band begins to play,
And people wanting to buy goods cheap
Had better not stay away!

If the OCR transcription is to be trusted, a notice in the July 5, 1899 Pittsburgh Press similarly reads:

The Elephant Now Moves Round,
The band begins to play,
The Elks Street Fair is coming
And don't you stay away

Such references continued to made into the twentieth century.

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In his 1905 novel The Lady Navigators: And Incidentally the Man with the Nubble Brow, Edward Noble is the first I can find to pair the chorus with limericks (and in a maritime context as well, though not a chanteying one). In the novel, the ship's crew make a game of trading limericks (of insufficient quality and sufficient rarity to perhaps stem from the author's own experience—Noble, a contemporary and acquaintance of Conrad, spent 16 years at sea) to this delightful permutation of the chorus:

Oh, the elephant walks around
And the band begins to play
The silly young geese go quack, quack, quack—
And the donkey shouts haw-hay.

The sequence begins on page 101.

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This is just a cursory bit of research. I expect a more comprehensive search might yield more further results.

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Bonus reference to the 1927 death of a Captain Eric Atterling of the Swedish motorship Santos in the July 11 edition of the Oakland Tribune. Perhaps Hugill's informant?