Then she got it wrong. Who in their right mind would write a song and devise a virtual nonsense line which doesn't rhyme with the word it's devised to rhyme with?

Let's examine that shall we?
Sing laddie ae and sing laddie oh,
The plooman laddies are a' the go.

That's what's known as a 'perfect' rhyme, as you are all no doubt aware. Why would any songwriter, even the most unlettered, or untutored, transpose the line to read?

Sing laddie oh, and sing lady ae.
The plooman laddies are a' the go.

No, no, Lucy just got it wrong, and others have followed her slavishly ever since, but wrong is wrong, no matter what the scource.