There is no issue with the surname Edwards. It has nothing to do with Mary Ann being deaf and dumb. The children were illegitimate and their births were registered in the surname of their mother apart from Fanny (if this is the right certificate) where Mary Ann bluffed it out with the registrar and had her 'legitimately' registered. I have and know of many examples (as late as the 1930s) where other mother's similarly had their children registered as legitimate offspring.
Fanny Bishop/Edwards has three different birthplaces on the censuses
Wisborough Green
Egdean
Stopham
I would think the most likely place for her baptism was Wisborough Green, but Petworth or Egdean are also candidates - Stopham seems a 1901 red herring.
James William is found on the 1881 census - birth Waltham (no James William Bishop birth registered in Sussex, only one William James Bishop and that was in Brighton). By the birthplaces of his siblings James was brought up in the Petworth area. There appear to be no other Mary Anns born Sutton of the same age who were deaf and dumb from birth.
Short of waiting for the release of the 1911 census to see what birthplace William James Edwards gives then, I'm not sure how proveable it is that his was the baptism in Cold Waltham. If both men were illegitimate the same problems apply - sometimes missing birth registrations, appearing in censuses in different surnames from the one used as an adult e.g. because mother's surname used on earlier censuses and then used father's when an adult, enumerated in a stepfather's surname but changed to either mother's surname or father's when adult. The surname you are enumerated in on a census does not mean that is the surname you were known as by your neighbours - the one you used on a daily basis.
1891 census RG12 832 folio 127
Egdean Sussex
William Bishop 17 Egdean, Sussex, Lodger Ag lab
Nobody with the surname Bishop born Egdean (or anywhere nearby) within 10 years of a circa 1873 birth date on the 1881 or 1901 censuses.
Regards
Valda