I found the whole thing fairly quick and painless on the NHS.
What I would say is don't be afraid to overexplain things. And if they give you a questionnaire where you can't choose a response because none are exactly accurate so you get very stressed, tell them that. Wanting to put across the exact truth and not misrepresent yourself at all is an autistic trait, so don't be afraid to go with any impulse to do it.
Allistic people overall tend more towards wanting to hear something that's straightforward and easy to understand, rather than hear what's true, and learning to give them that - which usually feels like bullshitting at best - is part of masking.
So take your time to answer questions - slow processing of conversation is also typical of autism, so again, you won't be misrepresenting yourself by going as slowly as you need to - and do correct whoever's referring/assessing you if they say something about you that's wrong. Similarly, you can bring up anything about the process so far that's bothered you.
I made a long list of everything I could think of that could be related, and took that with me. My psychiatrist offered to read it, and I declined just because it was written in a way designed to prompt me rather than to be read by someone else. But that means that handing over a list for them to read may be an option. Do make a note of even tiny things you think could be relevant, because most diagnoses of anything are cumulative.
I also mentioned reasons I'd previously dismissed the possibility of being autistic. For example, I'm hugely touch-seeking, and I'd only ever heard of autistic people being touch-averse. But it turns out that being touch-seeking is also an autistic trait; it is often the case that the opposite ends of each behavioural/experience scale are the autistic ways of being, and the middle that's not. E.g. hypo- and hyperempathy are both autistic.
And the eye contact thing seems to be a big deal. I don't mind it, I just don't understand it or know how much/little to make, and that was accepted as being a very significant symptom, so an aversion should count very much in favour of you being diagnosed.
You could also mention feeling such a strong need for a label for what's going on with you. Because I mentioned that to my GP as probably being a trait in itself - that I like to have things categorised and sorted out even if it's not necessary - and she agreed.
And I have found it helps. I've been surprised by how accepting people are once they know that that's why I am the way I am, actually. I didn't expect any improvement, in fact I expected more prejudice, not less.
But it's like I was getting all the prejudice anyway, because somehow they could already tell. There's been research that suggests that allistic people often form a negative opinion of us based even on so little as a glance at a static image, so that's absolutely possible; many have a kind of spidey sense that detects us. So being able to tell people I'm autistic has only improved matters.
Best of luck, whichever road you go down.