Also elusive, can't see any definite sign of John Edward after his birth. His birth has a mmn, Spellman, but it appears to be the only one and no obvious marriage. Impossible to connect him to William with any certainty.
It may be certificates are the only option (marriage then birth) but if William fibs to cover his possible illegitimacy then even then that might not help. If the dob of the Peterborough birth certificate is 15 Aug then that would be something concrete.