In a practical sense and as it stands, this is often the route trans people have to take:
It means many hours of someone going through their entire life history and specifically their personal issues where that persons relationship to their birth assigned gender/body has massively conflicted, to multiple doctors and psychiatrists. Then if someone passes the initial psych stuff, they get accepted on a care pathway where hormones are provided officially around 6-12 months into transition, but many have already self-medicated before that.
Either having entered a GIC (Gender Identity Clinic), or before they do so, a person renounces their previous name and identity and changes this through stat dec/deed poll, and then lives as their chosen/self-identified gender for at least 2 years (usually with work, etc. also providing supporting evidence of this being the case). After that, if that persons wants sex surgery, they then go through the entire life history stuff with another psychiatrist to get onward referrals.
To get a birth certificate amended, as mentioned above, also requires all of this to be applied for through a process with the GRP (Gender Recognition Panel) where they review all of the history and whether to grant a GRC (which allows the new birth certificate to be made). This does not have requirements for someone to have had surgery but needs further documented evidence of all the above.
All in all, it's a pretty long set of hoops to jump through and which weeds out 'chancers', even if not failsafe.