+1 to Jeatyn's comments about changing your name beforehand speeding things up. My Gender Therapist was impressed that I'd taken this step before my first appointment; she said it demonstrated my commitment to the process. In fact, she counted the official start of my RLE as being the date on which I officially changed my name (although I committed to my social transition six months previously) so it really did move things along for me. If you have no reason to wait, why not change it as soon as possible.
A couple of other points:
I went the lawyer route for my Deed Poll because of my Asperger's: there's nothing that messes with my anxiety more than feeling that I've not done something the 'correct' way. Getting a lawyer to notarise the Deed Poll adds an extra air of authority to the document, which can help when dealing with certain officious bureaucrats. However, most of the businesses I've contacted were happy to change my name & title merely on the strength of a phone call, even with my ridiculously girly voice: it was only the more formal organisations (such as banks) that insisted on Certified Copies or seeing an original document.
Please be aware: under Common Law, you don't need anything in writing to change your name and/or title. You can call yourself anything you like (apart from certain professional and noble titles) at any time without drawing up any formal documentation. The benefit to getting a Deed Poll is that you then have an official-looking piece of paper that you can show to bureaucrats & officials, because many of them aren't satisfied with just taking your word for things.

Also, Leo, I thought you might want to know that the wait between GPs and the community psychiatrists is very quick for most people... but unfortunately the big wait is between the community psychiatrist and the GIC. The reason for this is two-fold: 1) GICs tend to have long waiting lists; and 2) they have to apply for Exceptional Funding for your case before they can see you, because GD isn't routinely funded. I saw my community psychiatrist exactly three weeks after seeing my GP, but it took another 8 months for me to get into the GIC.
So... when you see your community shrink, ask him which GICs are available in your area (generally speaking, they'll have funding arrangements with certain GICs only) then phone each of those to find out what their waiting times are before choosing one. I made the mistake of choosing the more famous one because my shrink didn't know about the massive difference in waiting times for each clinic.
And yes, a letter from your doctor saying you've been diagnosed with GD will enable you to change your gender marker with the DVLA & passport people... but this will almost certainly not come from your GP. At least, not until you've been assessed by the GIC and formally diagnosed with GD. My letter came from my GIC; my GP was not willing to write anything until I'd been formally assessed. You might be lucky & have a sympathetic GP, but don't be surprised if they tell you you have to wait until you've seen the GIC first.
Oh, and if anyone's curious, I'm also happy to let you know how to get a new NHS number once you've changed your name. It's a bit of a palaver but please let me know if you want more info. (You'll need a new number because NHS numbers are gender-specific, so your GP won't be able to change your title on their system without a new number - and you
don't want them to call for "Miss Joe Bloggs").