If you have a passport, you may wish to update this with your new name and correct gender. The process is slightly more involved than updating your drivers licence, and you will need a handful of documents before you begin.
To start with, you will need either a Gender Recognition Certificate or a deed poll, a letter from a doctor or medical consultant stating that your gender change is likely to be permanent, and evidence of you using your new name. You can find out how to obtain a Gender Recognition Certificate here.
The letter from a medical professional can be from your GP, or from somebody at a Gender Dysphoria Clinic, such as your lead professional. They must be medical staff and not administrative. The wording of the letter is important here - it must state that the change of gender is “likely to be permanent”.
The evidence of your new name can be any official paperwork that includes your name, such as a utility bill, payslip, or bank statement.
Once you have all of these documents, you will need to fill out a form, which you can find here or you can visit a Post Office and take a physical form. You will need to send all of the documents to them, along with a passport photo (if you are updating that), your old passport, and a cheque with the appropriate fee.
It is important to note that currently the only gender options available are male and female. There is not yet a third option, or 'X'.