"A roaring pair of bagpipes, made from the green willow"

Bellows have two plates, called the top plate and the bottom plate. Hence a "pair" of bellows (or bagpipes). The word "bellows" is singular, but has a plural form. Rather like "scissors" - also kingular, but which has a plural form.

Any wood might have be used to make the bellows plates, but a willow wand was needed as a rib to keep the leather in tension.

Hence, "green willow".

See this YouTube video on making old bellows:

https://www.youtube.com/watch?v=xmDCZKh5wkE

The word "cole" is of Middle English origin, meaning "swarthy", "coal-black" or "charcoal". This would seem to add weight to its origin as a blacksmith's song.

As to possible sexual connotations, lots of old song lyrics include a reference to a "lusty smith".

One version of the song includes this verse:

"Here's a health to a pretty maid,
The kind I love best,
Who kindles a fire,
All in her own breast."

Which adds an implied meaning to the chorus which follows it ...