England and Wales: the certificate stays as it was#
A birth certificate is a certified copy of the entry made when the birth was registered, and GOV.UK's guidance on correcting a registration says a correction can only put right something that was wrong at the time; it cannot add new information, such as a change of name made later. So an adult who changes their name keeps the same birth certificate.
That is not a problem, because the two documents are read together. The birth certificate proves the name you were given; the deed poll proves you gave it up and adopted another; and as long as the former name on the deed matches the certificate exactly, spelling and order included, the chain is unbroken. It is the pairing every organisation is checking when it asks for both.
A baby, within the first year
The name a baby was registered with can be changed, or added if none was given, within 12 months of the registration. The register office adds the new name to the entry without erasing the original, on a certificate from the parents, or from whoever performed a baptism if the name was given then. After 12 months, a child's name is changed by deed poll made by a parent with the agreement of everyone who has parental responsibility, and the certificate stays as registered.
Corrections
A genuine mistake in the original entry, such as a misspelling recorded at the time, can be corrected on application to the register office or the General Register Office, with evidence and a fee. That is a correction of the record, not a change of name.
Scotland: the change can be recorded#
If you were born or adopted in Scotland, National Records of Scotland can record a change of name against your birth entry, and new copies of the certificate then show your new name and your previous one. NRS says it costs £40, plus £10 for each additional family member applying at the same time, and that the service cannot be used for a change on marriage, divorce or civil partnership. It is optional: HM Passport Office and DVLA accept a deed poll wherever you were born. Changing your name in Scotland has the detail.
Northern Ireland: the same idea#
If your birth was registered in Northern Ireland, the General Register Office for Northern Ireland can record a change of forename or surname against the entry on application, and nidirect lists a deed poll as another way to change your name there. New certificates then show the change.
Born outside the UK#
A birth certificate is issued under the rules of the country where the birth was registered, so only that country's registry can say whether it records a change of name. In the UK your deed poll is the document that proves the change, shown alongside the foreign certificate.
