The caribous could easily be a coincidence, since "-oo" is a useful rhyme and suggested by the Canadian milieu. (See what I mean? It's even happening now!)

Another very real possibility is that Coward was familiar with the poem, even if he didn't write it.

The earliest printed reference I find, for what it's worth, came in 1941. No text, of course. The psychologist H. J. Eysenck wrote decades later that he'd "learned it at student parties at Exeter College, London, about 1932."

It was widely recited during WWII, apparently in the RAF especially.
Perhaps its direct inspiration was the bawdy parody of Service's "Shooting of Dan McGrew," but now I'm just guessing.

The limited information available suggests that "Eskimo Nell" has not been very well known in the U.S. That too may imply a British rather than a Canadian origin.