Hi
Good news and bad news I am afraid,with the details you have provided I have found referances in the 1851 census to both the National School and the name Jefferies both in the same series of folio numbers, the bad news is that the relavent census pages are in such a poor condition that I cannot read them. I would advise you contact Cheshire Records office at this link,
https://www.cheshire.gov.uk/recoff/eshop/genhist.htmgive them the details and request a copy of the folio.
quote reference HO107-2156-FOLIO 769
thier copy should be a lot better than mine.
I do not not know anything about the National school apart from the fact it existed in Wellington Road Stockport, in the 1851 census between folios 742 and 774, and would have been one of the many private schools that existed prior to compulsary education.
good luck
peterbennett