Variants started appearing immediately, as in the following publications. I have boldfaced the differences.

The Scots Nightingale: Or, Edinburgh Vocal Miscellany. A New and Select Collection of the Best Scots and English Songs; and a Great Number of Valuable Originals… (Edinburgh: James Murray, 1779), page 150.

A NEW SONG.
Tune, A hunting we will go.

[1] Behold upon the swelling seas,
  With streaming pendants gay,
Our gallant ships invite the waves,—Our gallant, &c.
  While glory leads the way.
    And a cruising we will go,—oho, oho, oho,
    And a cruising we will go,—oho, oho, oho,
      And cruising we will go,—o-oho,
    And a cruising we will go.

[2] Ye beauteous maids, your smiles bestow,
  For if you prove unkind,
How can we hope to beat the foe?—How can, &c.
  We leave our hearts behind.
    When a cruising, &c.

[3] See Hardy's flag once more display'd.
  Upon the deck he stands.
Britannia's [or “Old England’s”]* glory ne'er can fade,
  Or tarnish in his hands.
    So a cruising, &c.

[4] Be Britain to herself but true,
  To France defiance hurl’d.
Give peace, America, with you,
  And war with all the world.
    And a cruising, &c.

* This variant appears in:
St. Cecilia: or, the Lady's and Gentleman's harmonious companion: …, (Edinburgh: C. Wilson, 1779), page 113.

This song also appears in:

The Universal Scots Songster: A New Collection of the Most Celebrated Songs in Three Parts, Part 1 (Edinburgh: Printed and sold by all the Booksellers in Great Britain, and Ireland, 1781) page 168.

The Roxburghe Ballads: Illustrating the Last Years of the Stuarts, Vol. 8, edited by J. Woodfall Ebsworth (Hertford: The Ballad Society, 1897), page 323. (Titled “The Cruisers”)

Sea Songs and Ballads, edited by Christopher Stone (Oxford: Clarendon Press, 1906), page 37.

Publications of the Navy Records Society, Volume 33: Naval Songs and Ballads, edited by C. H. Firth (Navy Records Society, 1908), page 247.

Transactions of the Shropshire Archaeological and Natural History Society, Series 3, Vol. 8, (Shrewsbury: Adnitt and Naunton, 1908), page 350.

Iron Men & Wooden Ships, Deep Sea Chanties, edited by Frank Shay (Garden City, N.Y.: Doubleday, Page & Co., 1924), page 8.