Rather than us jumping straight in with the answers, have you tried the following?
(As rosie said) GRO birth index (not the one at Ancestry) - should confirm the mother's maiden name for the 4 children you mention, and you should be able to get full copies (pdf) of the registrations for Ł2.50 each.
1851 and later censuses - to find out where William was born and confirm his age; also in case his parents happened to be staying with them
1841 census - you might find William with his parents, but it's not such an unusual name, and remember that relationships and exact birthplaces aren't given; if nothing else it might give you some possibles to work with
Then start looking for possible baptisms, which should name William's father and possibly his mother; if not C of E, they could have been non-conformist, in which case you might need further help.
C of E records for most of the North Riding are at Ancestry, but if you don't find them there, look at FamilySearch (free, but registration needed). The exact location will determine what records are available and on which site, so if you need more help, be sure to tell us that.
Good luck!
