Gender-neutral names (names that do not carry strong gendered associations) are increasingly chosen by people across a wide spectrum of reasons and identities. You might be non-binary or genderfluid, or you might simply have always preferred a name that does not announce a gender. Whatever your reason, you do not need to justify it to anyone. In the UK, you have the right to choose any name for any reason, and a deed poll is the legal instrument that makes your chosen name official.
Gender-neutral names such as Alex, Avery, Charlie, Dakota, Ellis, Finley, Harley, Jamie, Jordan, Morgan, Quinn, River, Robin, Rowan, Sage, Sam, Skylar, and Taylor are all in common use and are accepted by every UK organisation for passports, driving licences, bank accounts, and medical records. There is no list of "approved" gender-neutral names, and no organisation has the right to refuse to update your records because of the gender associations (or lack of them) in your chosen name.
A deed poll for a gender-neutral name change works identically to any other deed poll. You declare your current name and the name you intend to use going forward. You sign in front of two adult witnesses, and the document is immediately legally valid. You can then present it to any organisation to have your records updated. Most organisations process gender-neutral name changes as routine; they are concerned only with the deed poll, not with your reasons for choosing the name.
Changing your name does not automatically change your gender marker on official documents. If you also want to update a gender marker (from M or F to X, for example, an option available on UK passports since 2023), that is a separate request you can make to the Passport Office when you submit your deed poll for a name update. These are two distinct processes, and you do not need to do both at the same time.
You also have the option of changing your title to something gender-neutral such as Mx when you notify organisations of your name change. Your deed poll covers the name; the title request is made directly to each organisation. Most major UK institutions now recognise Mx.