Thank you Henry and Iains. Since you have taken all this trouble I will tell you what I can read so far.
Firstly we know the date that Joseph Taylor heard Brigg Fair. It was on or about the 5th of August, as the song says. The Travellers were camped for Brigg Fair, as explained above by Mrs. Hudson.
Secondly we know that the 'Pit' was their traditional camping ground, and it must have been suitable to push willow spars into the ground for the ridge tents, and also had good grazing and a ready water supply. This was not only for 'domestic' use but to water the horses, of which there would have been many, all for sale at the fair, and to soak the wheels of the carts or drays, in hot weather. Shrinkage is the enemy of cart wheels, and the term 'cutting and Shutting' comes from wheelwrights cutting down shrunken wheels.
Thirdly there must have been a source of food. Local wildlife but also shops within horse driving distance Likewise there must have been civilisation for hawking the usual items. (That's another thread!)
Fourthly we can estimate the year for the singing of Brigg Fair. Joseph Taylor was working but still living at home with his mother. With his birth in 1833 we can estimate the year 1851/2.
Fifthly We know the song was sung by a younger Gypsy to the 'Maria Marten' tune. The Red Barn Murder had taken place some thirty years previously, and the Lazarus tune may have been adapted for the song, in the late 1830s or early forties. Steve Gardham would know more.
In the sixth place we know that the song was also sung by John Deere who may have learned from Taylor.
Finally the song mentions the 5th of August, which is fair day, has a well known tune, and the two verses may have been a local composition.
So we are left with the balance of probabilities.
For my money either our 'Young Gypsy' or person of his acquaintance, may have composed the song especially for fair day. Perhaps there were more verses. Joseph Taylor sat round the fire, was likely to remember a song sung by a Gypsy closer to his own age.
The well known words have been reconstructed by Grainger, with verses from 'Low Down in the Broom'.
In conclusion it would be wonderful to pinpoint which family of Gypsies sang the song. So if the excellent Henry and Iains can pin point a 'Pit' with the required facilities mentioned, I will make a few phone calls to Gypsy Folk of my acquaintance and see what memories linger. We might strike lucky.
I think I have been fairly logical here but my mind is wide open to other views.