Azizi - I've never noticed any relationship between the Guy Fawkes rhyme and the Build A Bonfire one. The tunes are totally different and usage never overlapped.
Remember, Remember... was strictly associated with Bonfire Night (5 November) and the Gunpowder Plot.
Build a bonfire... tended to belong to the community singing of the school bus trip on a day out, along with other songs which only ever seemed to appear when sitting on the back seat of the bus.
Quack!
GtD.