My sense is that this song fits among the stevedore songs, "Stormalong," "Santiana", and "Fire Marengo." The lyrical ideas and choruses became shared between the variations of these themes, so they don't always make obvious sense.

The "gained the day" pattern was used earlier in this context, and in African-American singing contexts that people so inclined would connect to the cotton-stowing songs.

There's this, attributed to Black men in Georgia rowing a boat:

Gen'el Jackson gain de day—
Whaw, my kingdom, fire away,
Be gain de day in Floraday,
Whaw, my kingdom, fire away.

[Nathanson, Y.S. "Negro Minstrelsy - Ancient and Modern." _Putnam's Monthly_ 5(25) (Jan. 1855).]

Then there's this cotton-screwing chant, supposed to have been heard in 1838:

"Gin'ral Jackson gain'd the day;
   Fire the ringo, &c.
At New Orleans he won the day;
    Fire the ringo, fire away!"

[1859        Gosse, Phillip Henry. Letters from Alabama. London: Morgan and Chase.]

There is also this Black stevedores' song, in a work of fiction, but which I believe is based in first-hand experience:

"Gen'ral Jackson's a fightin'-man,
    Fire, my ringo, fire away;
He opened his forts, fired away,
    Fire, my ringo, fire away."

[1881        Kellogg, Elijah. A Strong Arm and a Mother's Blessing. Boston: Lee and Shepard.]

And finally there's this cotton-stowing excerpt, heard in 1845:

In New Orleans they say,
    Fire, maringo, fire away,
That General Jackson's gained the day,
   Fire, maringo, fire away!

[1896        Erskine, Charles. Twenty Years Before the Mast. Philadelphia: George W. Jacobs & Co.]

All are versions of a "Fire Marengo." They suggest to me that the pattern of "gained the day" was already there amongst African-American workers' songs. When the Mexican-American War came round, as it was these singers' habit to throw in topical lyrics, they could slip in Gen. Taylor.

So one of the historical versions of "Santianna", attributed to a ship in 1862, has these verses, with all the characters assembled:

General Taylor gained the day,
    Hurrah Santa Anna!
General Taylor gained the day
    All on the plains of Mexico.

He gained the day at Monterey,
    Hurrah Santa Anna!
He gained the day at Monterey,
    All on the plains of Mexico.

Santa Anna ran away,
    Hurrah Santa Anna!
He ran away from Monterey,
    All on the plains of Mexico.

General Jackson's at New Orleans,
    Hurrah Santa Anna!
General Jackson's at New Orleans,
    All on the plains of Mexico.

'Twas there he gave the British beans,
    Hurrah Santa Anna!
'Twas there he gave the British beans,
    All on the plains of Mexico.

[1904        Stacy, Rev. Thomas Hobbs. Fifty-Three Years Missionary to India. Boston: The Morning Star Publishing House.]

[The couplet in the last two verses, also mentioned in connection with other chanties, could have its origin in a minstrel song lyric:

Is dere any one here loves massa Jackson
Yes I's de nigga loves General Jackson
...

He thrashed the red coats at Orleans
He gib Packenham all sorts of beans]

There are at least two good versions of "Santiana" on record with General Taylor gaining the day, not Santa Anna. I think the "British supporters of Mexico" theory doesn't have much to support it. The sources point to the cotton-stowers making up or developing these songs. Just the fact that Santa Anna ends up gaining the day in many versions does not, in my opinion, point to a different group making up the song. Maybe some one switched the verse around, maybe someone made a mistake, or it was wishful thinking, but the song structure was already developed.

As for "General Taylor" being a distinct chanty, there is not a lot of evidence. There is the cotton stowing song in Nordhoff with "carry me along" (the similarity to Afro-American folk songs which has been pointed out). Then there is John Short's brief rendition -- in which a theme is barely developed. Seems to have more to do with eulogizing dead (i.e. similarly to Stormalong), perhaps, than following a narrative about Taylor specifically. Beyond this, if I'm not mistaken, we just have Hugill's. Who knows what he cooked up? Her certainly may have spun out verses to make it look like the theme was more cohesive. My opinion is that a bit of firmer narrative must have been woven around the song during the Revival post 1960s.