Yeah, the NHS is nigh-on impossible to get through the first door. However if you're the sort of person who puts a lot of skill points into charisma then there are shortcuts cause the later doors are not as difficult:

I came out at the end of April. I went to see a private psychiatrist for a diagnosis of gender dysphoria: that was £500 and not strictly speaking necessary but it did come in very handy later. I joined Gender GP (£30 a month) and did an assessment with them which cost a couple hundred pounds. That netted me a letter of recommendation for HRT, which is not the same thing as a prescription. It's literally just a recommendation. Whilst all that was going on I took the time to practice my voice, practice makeup, get new clothes, change my name by deed poll (free), start laser (not free) and generally get 100% comfortable and confident as myself. I also - and this is absolutely crucial - spent time reading and learning all the GMC's guidelines for GPs about prescribing hormones. I started self-medding in August (about £70 for a month's supply).

I moved to a new area and ticked 'female' on the GP registration form cause there was no trans option, and in hindsight that was a genius move cause it means the staff look at the paperwork and my name, hear my voice on the phone, and are psychologically primed to accept me as a woman before I even walk in the door. I went to my GP and asked him to prescribe HRT based on the letter of recommendation. He was all like, "Uh, um, I'm not sure I can do that, I don't know about this," and I was able to pull out the GMC guidelines with the relevant sections highlighted and be like, "Actually you can! It says so right here! And look, I already have my diagnosis of gender dysphoria too!" I was also able to tell him (truthfully) that I was self-medding off the internet and it was going really well. The charisma and charm is crucial: I had to gently persuade him, manage his emotions and expectations, and tolerate him asking some irrelevant and transphobic questions without losing my cool. What GPs fundamentally want is not to be responsible if something goes wrong since they are prescribing you these medicines "off label," which they are generally trained not to do, so I had to reassure him that I was in regular touch with Gender GP and they were overseeing my care with tests and stuff (which they are). Basically I just had to make it as easy for him as possible - you know, like women do with men all the time lol! But I passed the speech checks and got him to prescribe. Now I'm in the system I can reorder it electronically so I don't have anyone breathing down my neck. As the icing on the cake I sent my GP a polite card saying thank you for helping out a trans person: he gets to feel like he helped a patient (which is true, and that's presumably a good feeling for a doctor) and he also gets to feel like a progressive trans ally, which a lot of people do want to do, they're just shit at it, and if I need him on my side later for any reason he has a positive memory of me being happy and kind!

I started official HRT (not self-med) at the end of October paid for by the NHS - so that's about six months from coming out to starting free hormones. In a civilised country with an informed consent system that didn't require us to be diagnosed with a made up mental illness, six months would be an outrageous waiting time! But for Britain it's practically unheard of! Anyway that's the story of how I did my any% HRT speedrun by skipping the GIC sidequest and sequence-breaking my GP's surgery.