The six U.S. armored cruisers mentioned by Thomas were launched in 1906.

Four more - North Carolina, California, Montana, and South Dakota - were commissioned in 1908.

So the most likely date for Thomas's verses appears to be 1906-1907.

From the same period comes this information: At Yale in 1906 was sung a version of "that...song...reputed to have been sung by Uncle Sam’s tars on their cruise round the world [in 1907-1909]:

“Away, away with fife and drum!
Here we come, here we come!
We’re looking for something to put on the fritz,
The piratical class of nineteen six.”

(Edwin R. Embree, "History of the Class of 1906, Yale College" [1911])

None of this says much, about the obviously related "North Atlantic Squadron." Now associated with the Royal Canadian Navy, it seems just as likely to have begun in the Air Force, in light of its various air force allusions. The RCAF No. 10 (Army Co-operation) Squadron (redesignated "No. 400 Squadron" in 1941), is said to have been nicknamed the "North Atlantic Squadron," but possibly only after the song had become popular.

Hopkins avers that the song was "well established in the Canadian military" by the 1930s. Well, maybe.

The USN "Book of Navy Songs" (1926, 1930) has seven humorous but perfectly respectable stanzas of "The Old Destroyer Squadron," to the tune of "Away with Rum" or "The Armored Cruiser Squadron." In form these resemble Prentice Strong's 1912 Army National Guard version (above).

Niles, Moore, and Walgren's sometimes reliable "The Songs My Mother Never Taught Me" (1929) includes a single stanza of "The Armored Cruiser Squadron," to the tune of "Away With Rum," and "sung very often [during WW1] by the old-timers in the U.S. Navy:

"Away, away, with sword and drum,
Here we come, full or rum,
Looking for some one to put on the bum,
The Armored Cruiser Squadron."

The verse presumably came from co-author and USN veteran Douglas Moore
(later a prominent American composer.)

Niles et al. also give a "Northern Bombing Squadron," from Lieutenant Beauregard Sweeney, USN:

"Oh, the F 2 A and the H-S one,
The finest ships you ever did see,
Flew across the sea to be
In the Northern Bombing Squadron.

Away, away, with sword and drum,
Here we come, full of rum,
Looking for something to put on the bum,
The Naval Aviation."

Interestingly, the NBS never used F2 or HS-1 flying boats.

Hopkins's (again respectable) version of "The Old Destroyer Squadron"
is mostly made up of stanzas from the USN version. Clearly from WW2,   Hopkins's "West Atlantic Squadron" is equally tame. He mentions the existence of a WW1 "Heavy Cruiser Squadron" too, but offers no text or reference.

To "put someone or something on the bum" became a widely used idiom shortly before 1900.

No "North Atlantic Squadron" seems ever to have existed in the Canadian Navy. The USN "North Atlantic Squadron," however, existed from 1865-1902; in 1902 the name was changed to "North Atlantic Fleet."

My theory: Separated from the rest of the song, Harrigan and Brahms's "Away with Rum" chorus became widely popular in the late 19th century. A version of it (with "put on the bum" and "North Atlantic Squadron") evolved around 1900 and was widely sung in the USN by 1906.

Then,some time before 1939, the American "North Atlantic Squadron" chorus, possibly with ribald verses already, was adopted in Canada and vastly expanded.

The phrase "put on the bum" was easily parodied.