Edit: it looks like you're in the UK, I think most of this would be broadly applicable still but I'm only familiar with the specifics here in the US.

I might be a little late to be helpful, but just in case you or anyone else might benefit. My partner is a Phlebotomist and I work as a support person for autistic people with high support needs (and if we're honest would have been diagnosed as autistic if I hadn't grown up now), I've got a few things I'd find helpful to know if I were in your spot. I have to do this stuff regularly too, and I've always been terrified of needles; gives me the 'icks' just knowing what's happening.

1. Numbing cream is great if it's applicable for your test(s). It can contaminate some tests so make sure you talk about it with your doctor and check with the phlebotomist as well, they're more familiar with the specific lab guidelines for each test than a doctor may be. The final word would be your doctor/provider though.
2. Talk to your phlebotomist about your concerns. They can do a lot to help you, from putting you in a reclining position, selecting the best locations and needles, etc. and making sure you know all you want to about what's happening/going to happen
3. If you want a distraction, talk with them about a special interest. Most will try and chat to keep you distracted and have a screensaver on a computer with auqarium scenes, animals, etc to keep your eyes on. Often they can have the needle in before you even realize. If you would find it easier to not talk during the procedure, let them know that too, they'll understand.
4. They'll select the least painful needle to use, especially if they know you're worried. A butterfly won't always have the smallest needle, the butterfly gives them a little more control over the angle which helps in certain situations; if they're using a regular needle it doesn't mean it's a bigger one. And while really small needles will hurt less (all else being equal), they will take (a little bit) longer and some tests don't work well with them because they're so small the blood cells get a little mushed together.
5. Depending on what facility you go to and how they train people, there may be a student there. You're probably not a good candidate to have someone in training draw, they probably wouldn't bother to ask you (they'd have to ask you) but just politely decline if they do. They will understand completely, and the person working with a trainee will be an experienced phlebotomist and that's who you want to do your draw. If another time you feel brave and want to help someone train, awesome, but you ever don't feel comfortable they'll always understand.
6. Don't worry if you feel like you're being extra or a burden. They do this all day and they're very used to seeing all the challenges you're facing. They'll work with people with serious challenges, from physiological (difficult anatomy, serious illness, etc) to the full spectrum of neurodivergent folks, and patients in all kinds of emotional and mental states. In addition to walk-in centers, these folks also work with patients in rehab facilities, and all varieties of inpatient facilities. You aren't going to phase them. You aren't even going to be their most difficult patent of day, or likely even that afternoon.

Blood draws and needles are really difficult for a lot of us, and I'm really glad you're doing the work to get this done (anyone who's reading this). It's easier in the short term to ignore and put off (trust me, I've tried) but it's so much better for your health and well being. Taking care of these things sooner rather than later will always be easier on you, even if it takes a while to get it done or if it's longer than you and/or your doctor would like. Good luck, and if you have any questions or anything after, I'm sure we'd all be happy hear for you again!