I don't personally own a copy of the liner notes from Lloyd's recording, so I will rely on the quotation on the Mainly Norfolk website. They quote:
'Like Bold Riley O, this tune (a Dorian one) was brought to Liverpool from the West Indies where a variant of it had served as a challenging stick-fight song. Among the vessels that adopted the tune as a shanty for heavy hauling were those running up the coast of Chile. Oldtime sailors, who had a high regard for Valparaiso women, pronounced the name of the country to rhyme with “versatile”.'
I've cast my eyes on a lot of written material related to chanties, and I haven't yet encountered "Bring 'Em Down" as anything brought by Liverpool sailors. Would love to see it, if so. It seems we are left to take Lloyd's word and imagine he obtained it from an insider oral source? However, saying it is from "the West Indies" hints at Jekyll's book as the possible source. And maybe someone got digging songs (certainly a Jamaican thing, but not limited to Jamaica) mixed up with stick fighting songs (a fairly distinct Trinidad thing, as far as I know). (The distinction may be glossed over by saying "West Indies".)
"Bring Dem Come" in Jamaican dialect means "let them come," as in "if they want to challenge us, go ahead, let them come and challenge." (Maybe the call to challenge was how the "stick fighting" confusion got there.) In Lloyd's deepwater sailor presentation, however, it's as if "Bring 'em down" is supposed to refer to someone hauling on a downhaul or some other line in a downward direction. We can imagine, again, that some hypothetical deepwater sailors were the ones to mis-translate the Jamaican dialect and give "bring" a new meaning and turn "come" into "down." But I see so reason, so far, to do that.
Irksomely, Lloyd's liner note begins by saying the tune is in Dorian mode, which it is not in either version. The folk song collectors of Cecil Sharp's generation put a lot of stock in noting modes, which has some rationale based on what they (and people like Bartok) were trying to do. But it serves no function here, rather seeming more like something Lloyd does to imitate the style of the Folk Song Society.
Again, without other information, it seems like Lloyd wanted to insert what would be stereotypical sounding deepwater sailor verses—and thus the narrative of "brought to Liverpool" takes shape.
If it comes from Jekyll, then there are some interesting choices. Jekyll marks it as "Allegro," which this is not. More importantly, Jamaican digging songs are (at least in recordings) regular in meter/tempo rather than loose.