A deed poll is available to genderfluid people in exactly the same way as to anyone else. You do not need a diagnosis, a Gender Recognition Certificate, a referral letter, or any medical documentation to change your name. The UK name change process is entirely self-administered: you declare your current name, declare the name you intend to use, sign in front of two adult witnesses, and the document is legally valid. Your gender expression is irrelevant to the process.
Some genderfluid people use a name that is flexible or works across different gender expressions — something gender-neutral like Alex, River, or Morgan. Others have a name they predominantly use and want to formalise it legally. Whatever your situation, a deed poll gives you a legally recognised document that all UK organisations are required to process. You do not need to explain your gender identity to any witness or organisation as part of the name change.
It is worth knowing that a deed poll changes your name — not your gender marker on official documents. Passports, driving licences, and other documents have a separate gender field (typically M or F) which requires a different process to change. If you do not want a binary gender marker, some organisations such as the Passport Office now offer limited options for non-binary markers. A name change and a gender marker change are entirely separate steps, and there is no requirement to do both.