The only way to go through the Richardsons on CWGC Website is to open any that would work out as a correct birth age and possible what area his regiment was from, some of the details do include parents names and or spouse.
I have had to go through pages to find what i want, it is not brilliant unless you have more information.
That is what i did for the Wager entry, though there were only 6 of them, i picked what i thought might match.
It is possible that Sophia had been widowed whilst pregnant and possible she had William before or just after her marriage to Adam Richardson.
You might like to try a website that does a service record I know A*****ry does some, on that document it might have parents and spouse, not sure on costs though and yes it is very expensive hobby, depends on what you want and what country you live in.
Claire