The political connection appears to be unfounded.
"The Worme of Lambton" is given as a tale, not a song, in Sir Cuthbert Sharpe's 'The Bishoprick Garland', 1834 (reprinted 1969). Four verses are given there of a fragment of an old ballad, which may have been connected to the story.
Sir John Lambton was evidently a Knight of Rhodes.
Another Northumberland worm (serpent)is "The Laidley Worm of Spindelston Heughs", which Child gave as an appendix to #34, "Kemp Owyne".