Sorry, it's taken me ages to come back to the thread, so have only just seen your question mj. Thank you for the welcome. 
I asked for a referral for adult assessment last year, having read a lot about how differently AS presents in girls after my DD was diagnosed with AS. So I guess my assessment story starts with her, really. I did years of to-ing and fro-ing in my head about whether to get my DD assessed, as in some ways she fitted the picture but in others she didn't. I took her out of school when she was 8, because she wasn't happy there, and decided to try home edding her. I think that made the diagnostic procedure easier because I've read so many accounts of having to fight for a diagnosis in the face of schools saying 'oh we don't see any problem' or 'she's fine at school'. I started reading more about autism/AS, and it was reading Tony Attwood's Complete Guide to AS, and how it presents differently in girls, that finally prompted me to ask for a referral. I wrote a list of concerns (including the aspects that I felt weren't typical of autism), and took them to the GP. It was whilst writing that list that the penny dropped a bit for me as well, as I thought 'oh I do/did that too!'.
I was lucky, I think, in happening to see a trainee GP (he'd just started and was giving all his patients feedback forms to fill out after the consultation) who didn't know quite what to do so just did as he was told. 
When I asked for the referral he said that her school should deal with that, and was totally flummoxed when I said she was home ed. he wasn't British, hadn't heard of HE and couldn't believe you were allowed to do such a thing! He hummed and hahed and in the end I said firmly 'I want a paediatric referral to X hospital' and he made the referral and included the list of concerns I'd printed off. After a couple of months we saw a fantastic developmental paed, who said that in her opinion DD has AS. I was afraid I was going to get laughed out of the place, especially as when the paed came out to get us, DD marched up to her, made perfect eye contact and shook her hand. She didn't do any of the AQ tests or anything, just had a long talk with her and me and used her clinical expertise (she told me afterwards that the Attwood book is 'their bible' when it comes to diagnosing in girls. She was really skilled and knew just the right questions to ask DD to get to the relevant stuff. Her diagnosis was confirmed by the SALT and the clinical psychologist over the next six months or so, and her official tripartite diagnosis is AS.
As I did more reading about it in the wake of that, the more it dawned on me that I had many of the same traits and might also be on the spectrum. I mentioned it to one GP who was quite disparaging and said that even if I was, a 'label' wouldn't make any difference. I left it for quite a while after that, but in a bad spell last year (having come off antidepressants), I felt I needed to know one way or the other. I asked my lovely female GP, explaining why I wanted to know, and she referred me to the adult assessment service. After a couple of months I got a letter to phone the service, which I did, and was put on te waiting list. They warned me it would be at least a year, but said I could go on the cancellation list if I liked. That was in the October, and a cancellation came up in the January! 
I was seen by the clinical psychologist autism specialist, who then wrote his report and discussed it with his boss, a clinical psychiatrist, who looked at whether it met the criteria for diagnosis. I was sent some forms by email to complete before the appointment; some tick box forms like the AQ test, a sensory profiling one, and a longer one that asked open-ended questions. For that one I wrote 8000 words 
and when I was panicking to DP that I'd have to edit it, he said 'no I think he can gauge a lot for the fact that you wrote all that'.
The appointment was scheduled for two hours, but I was there for almost four and a half hours. I only realised when I came out and it was dark! I was so nervous as the start I could barely speak, but he was absolutely lovely and really put me at my ease. He told me that they use the DSM-V criteria in this county (Surrey), which has incorporated AS into the general diagnosis of ASD. So my letter/report says ASD, but when he phoned me to tell me the result, he said that 'in old money', it would have been an Aspergers diagnosis.
In the appointment we talked about a huge range of things, but within the framework of the report, and he wrote down everything I said. They would have liked me to take my mum along, but she lives a long way away so I had written down her answers to the questions about what I was like as a child, andgave him her number to phone her if he wanted (he didn't do so). He told me towards the end of the assessment that in terms of meeting their tick-box criteria, I was right on the borderline of being on the spectrum or not, but that in his clinical experience, particularly of how women/girls present differently, he would put me on the spectrum - he drew me a picture to illustrate all this. He said he's seen more women presenting for assessment in the last few years, and said that I reminded him of four other women he'd seen recently... 'very different in terms of your personalities, but presenting with the same expressions of the condition'. He told me at the end that in his opinion, I had AS, but that his boss was very particular about all the criteria being met. I was expecting not to get the diagnosis, so was absolutely gobsmacked when he rung me a week or two later to say yes.
It took me ages to accept it - I thought he must be wrong (despite years of clinical experience! ), or that I'd misled him, or selectively remembered things (I rang my mum and read her all the stuff I'd written sbout my childhood, to make sure I hadn't given a false picture - she said it was all accurate). I still don't believe it sometimes, or think I'm an imposter. I've had a lifetime of feeling a failure for not being able to cope with things that other people just seem to get on with, to deal with. The reason I wanted the assessment was to 'settle, once and for all', whether I'm just fucking useless, or whether there's a reason for how I am. Of course it hasn't settled it once and fo all, and I still beat myself up a lot, but slowly, I think I'm learning to accept it, accept me, and forgive myself.
Wow that was really long, sorry! You can see how I wrote that 8000 word form, can't you. Hope that answered your question.