"Old Colony Times" (ca 1800?) is reproduced in S. Foster Damon's Series of Old American Songs (Brown University Library, 1936, No. 6; with tune), with this note:

This ballad may have preceded even the first attempts at colonization in our country; but the opening lines of this version, sung from Maine to Georgia, and at least as far as Nebraska, were probably shaped about 1800, when "old colony times" began to seem very remote.
When John Lothrop Motley studies at Göttingen in 1832, he taught this song, one of his favorites, to his fellow student, Bismarck. Over fifty years later, in a speech before the Reichstag on February 6, 1888, Bismarck quoted "Old Colony Times", which he had learned from "his dear deceased friend", Motley. (Orie William Long: Literary Pioneers, Cambridge, 1935). This song is sung by the archbishop in Agnes Repplier's In Our Convent Days; and in Hardy's Under the Greenwood Tree (Pt IV, ch 2) it is also to be found, beginning, however, "When Arthur first his court began".
Where sheep were raised, the "miller" ran a carding mill (see Flanders & Brown's Vermont Folk Songs, "The Farmer's Three Sons"). In the colleges, the tune was once much used for less familiar texts (Journal Am. Folk Lore XXIX, 167; see also XXX, 350; XLV, 47).

    OLD COLONY TIMES

    1.
In good Old Colony times
When we were under the king
Three roguish chaps fell into mishaps,
Because they could not sing

    Because they could not sing
    Because they could not sing
    Three roguish chaps fell into mishaps,
    Because they could not sing.

    2.
The first he was a Miller,
And the second he was a Weaver,
And the third he was a little Tailor,
Three roguish chaps together.

    3.
Now the Miller he stole corn
And the Weaver he stole yarn
And the little Tailor stole broadcloth for
To keep these three thieves warm.

    4.
The Miller got drown'd in his dam
The Weaver got hung in his yarn
And the devil clapp'd his paw on the little Tailor,
With the broadcloth under his arm.

This version is copied in Margaret Bradford Boni's Fireside Book of Favorite American Songs (Simon and Schuster, 1952, pp. 252-253).

Later editions are at American Memory:

Good old colony times, and Bonny boat. ( Sold, wholesale and retail, by L. Deming, No. 62, Hanover Street, 2d door from Friend street, Boston. [n. d.]) [text only]

Old Colony times (Cincinnati: Church & Co., John, 1878) [sheet music]

"The Noble Acts Newly Found, of Arthur of the Table Round" (To the Tune of Flying Fame) is at Bodleian Library Broadside Ballads.

Printers: Coles, F. (London); Vere, T. (London); Wright, J. (London); Clarke, J. (London)
Date: between 1674 and 1679
Imprint: Printed for F. Coles, T. Vere, J. Wright, and J. Clarke
Illus. Ballads on sheet: 2   
Copies: Wood 401(61)
Ballads: 1. The jolly pinder of Wakefield: with Robin Hood, Scarlet, and Iohn ("In Wakefield there lives a jolly pinder ...")
Subject: Clergy; Robin Hood$qlegendary character
2. The noble acts newly found, of Arthur of the table round ("When Arthur first in court began ...")
Author: Deloney, Thomas
To the tune of: Flying fame
Subject: Chivalry; Arthur$qlegendary figure

Two later parodies of "When Arthur first in court began" are at Bruce Olson's Roots of Folk site (Click here).