Thanks, Derek, for the wonderful information. I will say that, following my post, I was able to find Joshua Oldfield in the 1871 census. Apparently, while his family was living Snenton in Nottingham, Joshua was living in Arnold in Nottingham. He was a boarder there and working as a botanist. Not sure about what that means....he was a silk glove hand in 1841, and a Silk Framework Knitter in 1851 and 1861. (I haven't found him in 1861, but his wife, Jane, is listed as being a "wife of a FWK," which I took as being a Frame Work Knitter.)
I will say that the 1871 census has Joshua being born in Chesterfield, but I'm not sure if there is a major difference between that and Brampton, geographically speaking.
The children of Thomas Oldfield and Mary Hopkinson were all baptized into the Methodist Church; or, at least that's what it tells me on sites such as FamilySearch. It says "Methodist, Chesterfield, Derby, England." However, if this proves to be an incorrect connection then who knows to which sect he might have belonged!
Maybe this information might be able to help out a bit more. Thanks again!