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