Hi Emily.
I was already FT and had been on hormones for over a year, so seeing my GP dressed wasn't a problem for me. My GP basically looked at me and said, "Yeah, I'll refer you if you like." No questioning, no issues whatsoever.
Given that many GPs are still remarkably uninformed about transgender healthcare, it probably doesn't matter if you're dressed or not; go as you feel comfortable. I'm a huge believer in the idea that nobody in transition need torture themselves by jumping through humiliating hoops like dressing in public when you're not mentally ready. Go at your own pace, and presenting as male at your GP appointment is in no way a sign that you're not trans. Some GPs may refer you instantly, some may need to screw around for a while to figure out how to treat you and determine what services are available. Go in prepared: know what the GIC can offer, how they take referrals, etc. Print it all out if you like and take it in a folder to give to the doctor, along with some basic FAQs about transgender stuff. Assume the GP has never treated a trans patient before.
Good call on Northamptonshire. Supposedly by far the shortest of the waits, but it's still a slog to get that first appointment. Eighteen weeks from referral is the latest estimate (which compared to other GICs is positively lightspeed), but I strongly suspect that figure is very optimistic and the reality is somewhat slower; I'm still waiting. But even so, at least you're not getting screwed by Charing Cross.
I hope it all goes well.
