The oldest publication of the lyrics I can find is here:
From "The Waterman: A Ballad Opera" by Charles Dibdin, printed in The Monthly Miscellany, Volume 2 (London: R. Snagg, September, 1774), page 143:
[Note that the song appears without its own title.]
And did you not hear of a jolly young waterman, Who at Black-friar's bridge used for to ply; And he feather'd his oars with such skill and dexterity Winning each heart and delighting each eye; He look'd so neat and row'd so steadily, The maidens all flock'd in his boat so readily, And he eyed the young rogues with so charming an air, That this waterman ne'er was in want of a fare.
What sights of fine folks he oft row'd in his wherry, 'Twas clean'd out so nice and so painted withall: He always first oars when the fine city ladies, In a party to Ranelagh went or Vauxhall And oftentimes would they be giggling and leering, But 'twas all one to Tom, their gibing and jeering, For loving, or liking, he little did care, For this waterman ne'er was in want of a fare.
And yet, but to see how strangely things happen, As he row'd along, thinking of nothing at all, He was ply'd by a damsel so lovely and charming That she smiled, and so straitway in love he did fall; And wou'd this young damsel but banish his sorrow, He'd wed her to-night before to-morrow; And how should this waterman ever know care, When he's married, and never in want of a fare?
You can see musical notation for this song in Old Scotch Gems and Other Favourite Songs (Glasgow: David Bryce & Son, 1906), page 100, where the song is titled THE JOLLY YOUNG WATERMAN.