Which GIC are you going to?
As far as I know, most GICs will still insist on some RLE. A name change document won't be enough evidence of this on its own. You need more proof that you have been living full time. I took my deed poll, bank statements, bills, and a letter from my employer stating I'd been working as a woman from a certain date. I didn't need all of that in the end, they were happy with the deed poll, one bank statement and the letter from my employer. I had been full time for a total of seven months by the time of my first appointment.
My first session was basically a getting to know you. My coordinator asked about my background, both gender and family related. He also did a risk assessment to find out if I had any history of substance abuse, crime or violence, and to ascertain the likelihood of me hurting myself or anyone else. There is basically zero chance of you being prescribed hormones on your first visit, or even your second.
My second appointment was with both the coordinator and a psychiatrist. This was to get the diagnosis. I was then admitted onto the 'care pathway', and put on a waiting list for the hormone clinic (it took about three months).
My experience was that they didn't want to hold me up, and were keen to move me through the whole process as quickly as they could within the rules.