In most jurisdictions, probably including the UK, anyone can change their name for any non-fraudulent reason. I don't know the rules there, so you should confirm that, but it seems unlikely that they would require a diagnosis.
You can't change the gender marker on your birth certificate in the UK without a Gender Recognition Certificate, which requires a diagnosis of gender dysphoria. Basically, that means two years of RLE or being post-surgical, whichever comes first. This one I know, because I have a UK birth certificate.