I socially transitioned years and years before I got any hormones or surgery.
First I changed my name, you can just google for a statutory declaration template, you don't have to pay any lawyers or get anyone in particular to sign in - my brothers girlfriend witnessed mine. I then sent that off to anyone who had my details (my bank, anyone I pay bills to etc) with a letter saying I was transitioning and asked if it would be possible to change my gender marker at the same time.
Nobody had a problem with this, this included getting a passport and drivers license in my new name with the proper marker. You can also change the name for your national insurance details with a quick online form. (probably the same one hncpch posted)
The one last bit I still have left to sort out is my birth certificate, which is more complicated and requires a GRC and a bunch of paperwork.