Changing your name on your UK passport requires a deed poll as the primary supporting document. Once you have a signed deed poll, you apply for a new passport in your new name — this is treated as a change of name rather than a first-time application, so you will need to send your current passport alongside the deed poll. Applications can be made online or by post using the standard passport renewal process.
The Passport Office will not amend your existing passport — you will receive an entirely new one in your new name. The old passport is cancelled and returned to you (with holes punched through it) with your new one. Processing times for a standard online application are typically 10 weeks; a fast-track service (available for an additional fee) reduces this to around 1 week. During this time, your current passport is held by the Passport Office and you cannot travel internationally, so it is worth planning your application timing carefully.
One important note: you do not need to send your original signed deed poll to the Passport Office. A certified copy — a clear photocopy on which you have handwritten "I certify this is a true copy of the original deed poll" and signed it — is sufficient. Keep your original safe and work from certified copies for all organisations.