If someone is known by a different name from that on the birth certificate, which name should be used?

When banns are published, the “true name” of the persons who intend to marry must be used.  For the purpose of banns a person’s true name is not necessarily the full and exact Christian name given in baptism and the original surname. If a person has adopted a new first name and/or surname in such a way as to supersede his original name, and so that he or she is properly known by that name, it will be his or her true name for the purpose of banns.

For the marriage document, the clergy should follow the Guidance issued by the General Register Office in their Guidebook for the Clergy which is available at https://www.gov.uk/government/publications/guidance-for-the-clergy.  See p.20, section 4.