Becoming a British citizen through naturalisation is an opportunity to formally adopt a new name if you wish to. The UK Home Office allows applicants to take a new name at the point of naturalisation — if you notify them before your naturalisation ceremony, your chosen name can be recorded on your naturalisation certificate and subsequently on your British passport. This is a one-time opportunity to change your name as part of the naturalisation process without a separate deed poll, though a deed poll is the fallback if the timing does not work out.
If you decide to change your name after your naturalisation ceremony and certificate have been issued, a deed poll is the standard method. You sign the deed poll in the usual way, and from that point you can apply for your first British passport in your new name, as well as updating other UK records. Your naturalisation certificate will still show your name at the time of naturalisation, which is normal — the deed poll is the document that evidences the subsequent change.
For people changing their name as part of naturalisation who also want to update records with other organisations, the combination of a deed poll and your naturalisation certificate is a powerful evidence package. This service generates a correctly worded deed poll that is accepted by the Passport Office and all other UK organisations.