Kirsty
If this was the Cade Street Independent, then they were not anything in particular. It was apparently set up by George Gilbert (28-4-1741 to 23-3-1827).
"A wild, reckless and immoral soldier he was converted to 'Methodism' in 1766 and soon after became overseer of the workmen on the estate of his commanding officer, by now Lord Heathfield, at Heathfield Park. "
That was taken from Errey: Roots Branches and Leaves by Neil Errey, ISBN 0 9592787, Adelaide, Australia, 1982. I believe it is attributed to "Whe Worthies of Sussex" by M.A. Lower.
There is a lot more information in the book on George Gilbert and I guess the source, so I will not type it here. The HIC (Heathfield Independent Church) featuress heavily in the Errey Book and the author assumes/suggests it is the same one set up by George Gilbert. Some Errey women married Gilberts (unsure if related to George Gilbert). There are lots of mentions of the role of the HIC in Neil's book. It also mentions that the Errey's that came to Australia were not opposed to establishing their own, unconnected chapels.