No, you're completely wrong, I'm afraid. There's no law that says any sort of transition is "a fact", and nor is "transition" protected in the EqA.
The 2 relevant protected characteristics in the EqA are first, gender reassignment. This has a very specific definition (to do with undergoing/proposing to undergo a process, etc) and is not directly synonymous with "trans", "gender ID" or anything else. Moreover, the protection granted by the EqA is from being discriminated against, e.g. by being fired, refused a service or otherwise treated unfairly by employers/service providers compared to someone otherwise similar (including being the SAME sex!) but without this characteristic. It does NOT impose any requirement for people to be treated as the opposite sex in any way, including forcing others to speak about them in particular ways for example, it doesn't place any obligations/restrictions on anyone except employers/service providers, and it doesn't give any rights to opposite-sex facilities. This PC also only applies to the person who is undergoing/proposing the process.
The other PC that may apply (but may not, as it's not strictly been tested yet) is "Religion and belief", IF the belief in "gender identity"/"transition" was held to meet the criteria for being a protected belief. If so, it would similarly mean people could not be discriminated against by employers/service providers (as above) for holding that belief. As such, it would apply to anyone who believed in GI etc., not just those who actually identify as "trans" themselves. But again, it doesn't involve any requirement for other people to play along with this belief - they still have their own rights to free speech, their own beliefs etc., and anything that tries to impose or make them profess this belief if they don't hold it themselves (e.g. being required to call a man a "woman" at work, or be "educated" on trans nonsense as if it was fact) is likely to discriminate against them. Of course, it might be judged not to be a protected belief at all, in which case workplaces trying to impose it on people with "GC" (realist) beliefs would have even less of a leg to stand on!
As for the GRA, that only applies at all to the very small minority who've successfully applied for a GRC, and means that in certain limited scenarios, they are legally considered as the opposite sex. This is a "legal fiction", it doesn't make it "a fact", and indeed, the numerous restrictions as to what a GRC actually does make it quite clear that it was acknowledged that nobody actually changes sex. Again, it imposes absolutely NO requirements on individuals to pretend someone is the opposite sex, and as the Supreme Court clearly stated, it gives no right whatsoever to use opposite-sex facilities (cos again, they've NOT actually changed sex!). So that's no excuse either.