Tom Bolynn or Brian O'Lynn" is not American, but comes from a song sung by the fool in William Wager's play, "The Longer Thou Livest, the More Fool Thou Art," dated about 1565.
Randolph-Legman discuss the song in "Roll Me In Your Arms," vol. 1 of "Unprintable Ozark Folksongs and Folklore," pp. 155-157. William Allen White heard it in Kansas as "Barney O'Flynn."
Early versions appear in volumes by Peter Kennedy, "Folksongs of britain and Ireland," and Opie, "Oxford Dictionary of Nursery Rhymes."
I am sure Malcolm Douglas can give better information.

It survives in North America as a comic song, with many examples. Here is one from Randolph-Legman, 34B, with music, coll. in San Francisco from a British seaman:

BRINZI O'FLYNN

Oh Brinzi O'Flynn has the pox and the gleet,
And he stinks like a brothel down old Sackville Street,
With globules of mercury 'round his foreskin,
By God! I am dyin' says Brinzi O'Flynn.
Music given, "one of the finest airs in British balladry."
(Sackville Street is in Dublin, gleet = gonorrhea)

An American version from Joplin, MO, sung by Miss L. B.

JOHN BRINEY LINN

John Briney Linn, his wife an' her mother,
They all went out a-shittin' together;
Some of 'em shit needles an' others shit pins--
It's pretty sharp shittin', says John Briney Linn.

Variant sung by F. H., Berryville, Arkansas.

Bryan O'Linn and his wife and her mother,
They all went out a-shittin' together;
Some shit thick and some shit thin--
Wipe it up with a spoon, says Bryan O'Linn.

Music is given.
Randolph-Legman mention "Tumble O'Lynn's Farewell," which appeared in the Jour. American Folklore, 1990.
Versions have been played at the funerals of AIDS victims.

Stan Hugill has a shanty version in his "Sailing Ship Shanties" MS, according to Randolph-Legman. Has this been published yet?