It's interesting that you mention Barker's and Pottery in the same post.
I'm a descendant of a William Barker, my 5x Great Grandfather who was from Hook, Staffordshire, born 1766. I understand they were Potters, but have no idea if they are the same as the Potter Bros.
William moved to the North East of England in around 1787 when he married Sarah Hinton who was also a potter, as was her father John. All from Staffordshire but moved to the NE as potters.
Unfortunately the trail goes cold and I'm unsure about any further back. I have made an assumption
that the other two Barkers Edward and John who all married in Newcastle. I know that William settled first in Heworth south of the tyne, and then to Sunderland. Edward also did the same.
I cannot be 100% sure they were related, but Edward and William were both potters from Staffordshire, moved to the NE, married and ended up in Sunderland. A co-incidence perhaps, but interesting either way.
Maybe these Barkers were from a Branch connected ?