Steve, while "Thunder-Bomb" sounds like a perfect Gilbert & Sullivan warship, Les fron Hull is entirely correct about real bomb-vessels called "Thunder."

One, commanded by Capt. George Cocks, took part in the siege of Copenhagen in 1807. See, e.g., Sir Nicholas Harris Nicolas, The Dispatches and Letters of Vice Admiral Lord Viscount Nelson: With Notes, Vol. VI (London: Henry Colburn, 1846), p. 113. Admiral Nelson refers to the vessel as both "the Bomb-Vessel Thunder" and "the Thunder-Bomb," in the one dispatch in 1804.

And another reference to the ballad, from [W. N. Glascock] Naval Sketch-Book (London: For the Author, 1826) I , p.xiii:

"The late Tom Sheridan had a similarly brilliant conception in the comic song called 'Billy Taylor.' But Sheridan had imbibed more theatrical tact from his father; and by giving it an air of romance, completed the absurdity.--'Billy Taylor's' sweetheart is made lieutenant* of the 'Thunder-Bomb.' . . .

"*'When her captain come to diskiver
The glorious action what she'd done,
Then he made her first leaftenant [sic]
Of the gallant Thunder-Bomb.'"

If you can post the earliest version, I'll try to dig up that 18th C. "Calton Weaver" that misfired a few months back.