The origins of this are mysterious to me. Could it be one of those songs that circulated in men's clubs and other private male gatherings in the first half of the 20th century, and didn't enter folk tradition until later?

The earliest version I know is long before Nina and Frederick's: "Brother Noah," as sung by Sam Eskin c. 1947 for the Library of Congress. Is this closest to the original?? Its chorus does make more sense in an old testament context:

Hallelu, hallelu, hallelu, lu, lu.

But the song as sung by the U.S. Marines at an early date (1940s? 1950s?) uses this chorus: Tralaloo, tralaloo, tralaloo, loo, loo.

(The go-to-hell verse dates back much further, to World War I, as the chorus of an entirely different, much more literary glee-club-type song, "Old Noah," which begins "Old Noah built himself an ark, the good old Christian soul.")

Here's the version sung by Sam Eskin for the Library of Congress in 1947:

BROTHER NOAH

Brother Noah! Brother Noah!
Can I come into the ark of the Lord,
For it's getting' pretty dark, and it's rainin' very hard,

CHO
Hallelu,hallelu,
Hallelu, lu, lu.

No, you can't, sir, no, you can't, sir,
No you can't come into the ark of the Lord,
Though it's gettin' pretty dark, and it's rainin' very hard, CHO

Go to hell, sir! Go to hell, sir!
Go to hell with your damned old leaky old scow,
'Cause it ain't goin' to rain very hard anyhow! CHO

That's a lie, sir! That's a lie, sir!
That's a lie, for you know damn well it's a sprinkling now,
It's a-sprinklin' now, gonna rain like hell.

Note: The marine version differs most in the final verse:

Young feller! Young feller!
Thats a doggoned lie, for y' know damn well
That it's getting' very dark, goin' to rain like hell, CHO