noone in the wedding party signed the marriage certificate, they signed the marriage register, if the marriage was in an Anglican church it is the Vicar who filled in the certificate, the register can be altered by the church protocol being carried through to the bishop i believe. the vicar can amend the register only after then i think.
If it was a marriage in a register office, the superintendent registrar for marriages will have filled out the certificate and the register would have been signed by the parties. the superintendent registrar would be the person to ask in the first instance to alter the entry, it would probably be referred to the registrar generals office.
If it was a Catholic ceremony, the priest and a superintendent registrar would have made separate register entries and so ditto for each as above.
The first thing to do is to speak to the register office anyway. Bearing in mind that the couple really involved, i.e. the bride and groom are the people who informed the registrar/vicar/priest of the relevant family information, including occupations. The banns / wedding notice, would have been posted in the register office of both Parishes for up to 6 weeks before the ceremony when any errors in the information could have been amended.