Dunno why you hate this country sometimes; our passports are actually amongst the easiest in the world to change. We don't need to have undergone any medical transition in order to change the gender marker on our passports, and we certainly don't need a GRC either. Contrast this to some other countries where it is impossible to change the gender marker or name on your passport, or to others that require you to have been sterilised before you can change your documents. We've got it easy, believe me.

As Lisa posted, all you need is your old passport; your original Deed Poll (or Stat Dec) of name change; and a letter from either your GIC or your GP (GIC is better) stating that you've changed gender and you intend for this change to be permanent. Most GICs will give you this letter free of charge, but your GP may charge you for it.
Then you fill in the form - you can do it online, even whilst changing your gender on the passport - with your new name in the current name box, your deadname in the 'previous names' box, and then you submit the original documents together with a cover letter (which isn't essential, but it helps smooth things along) explaining that you need your name & gender marker changed as you're transitioning. You'll no doubt want a new photo, too - and this has to be countersigned as it is likely to be significantly different from your previous photo.
It's really simple to do, and it doesn't take any longer than a standard passport application.
BTW, most countries require proof of citizenship or entitlement to work if you apply for jobs. This isn't a British thing; in fact, it took a very long time for the British government to catch up with what the rest of the world has been doing for decades!