I used to live in the UK (for 29 years from birth) and I am a trans woman.
I can speak to you more if you'd like, but essentially what I recommend is the following:
1) Be prepared to wait. Get down to your GP and ask, nay demand (you are entitled to it), to a Gender Identity Clinic of your choice. This will get you on the waiting list for NHS care, but you will be waiting for a few years for it. Do not get referred to the Laurels in Exeter, even if you live nearby. You will wait forever for that clinic. I can't sugar coat that, so in the meantime...
2) Can you afford private help? If so, you can seek that. GenderCare or Gender GP (if they still exist) are your main options. GenderCare is on balance, probably better for GP acceptance. I have went through both and self medicated for reference.
When you see your GP for stage 1, ask if they will accept the shared care agreement between them and the private clinic. This means that the private clinic will provide a regimen for the GP to prescribe on the NHS while you wait to actually see the NHS. It will lower your costs quite a bit. You can Google Gender Care for contact details to get a copy of the shared care agreement to show your GP. Nail them down in agreement quickly so you can start getting down the private route. With GenderCare it could take four to ten months to start receiving hormones and cost maybe £600 to get started IF your GP is supportive.
3) If you cannot withstand the waits or the costs for either of these options, importing oestrogen and testosterone blockers into the UK for personal use is legal. This will provide you with a last resort backup that is both less time consuming and costly than the above, but it is stressful hoping your meds arrive and obviously it is less safe (though if you are healthy then it is not critically dangerous).
NHS guidelines are that GPs should give you a bridging prescription to tide you over til your GIC appointment if you are in danger due to low mood or self medicating due to the absurd wait times, buuuuuuuut these are not enforceable guidelines so most GPs will do their best to avoid helping in this way because they are cowards (used to assistant manage a GP practice, so I know this). It couldn't hurt to ask though, and you may have a nice GP.
I am typing on a phone, which is like walking through treacle, but if you want to discuss more you know where I am. And if any doctors give you shit or put up roadblocks, let me know because there may be obligations that we can cite back to them to put a fire under their ass.
Transitioning as a trans person is the greatest, and no election (no matter how sickening the result) should take that away from you. I know it's scary and I know it will be tough. I know the UK is a bit shit and a bit TERFy. But if you need this, it will be the right thing for you.
Best of luck.