Oh yes, get her a trial lesson somewhere near you and see how she gets on! It may indeed be right up her street from the description she’s given, and there are plenty of ND kids who participate in theatre. This makes a lot of sense to me because standing up in front of a giant crowd of people gives the average person fear, so it tracks that people who are fueled by it might have minds that work a little differently to average! It is a place that can celebrate all kinds of people.
Of course it’s not some perfect world either so consider contacting a school and giving them the same description: child has a special interest in musical theatre and is a gifted musician. She doesn’t have dance training and her coordination needs development, but she would like to give this a go. If there are any specific ASD issues you would want the teacher to keep in mind, you can mention that too and see how they respond. Some schools are great with it. DS goes to a programme that awards scholarships specifically for autistic children who want to perform, and teachers have specific guidance etc, so it can be a very supportive place. If it doesn’t sound supportive and rings alarm bells, then I would say move on to the next place and try there. You’re looking for somewhere with the attitude of dance and drama can help build her coordination, not a place that expects lots of dance technique when they walk in the door. (And it definitely can help build coordination! Many a happy dancer has struggled for marks in PE over the years!)
Unfortunately I don’t have northern knowledge for specific schools, but trial lessons are a very standard thing across the industry, so that’s what I would ask for. Some offer a free trial but most will offer a trial where you pay just for that day’s lesson without any obligation to continue. You shouldn’t be made to sign up for a full term without seeing if it will work for her. If you think she’ll need a few lessons to settle, discuss that with them in advance and see if they will accommodate. Half a term would be too much but two lessons totally reasonable IMO.
As far as favouritism, try to prep her that some kids will have been there a while and already in their patterns so yes, it may seem a little awkward at first but it should be better soon. And it shouldn’t be horrible - standard making new friends stuff fine, any sort of teasing not fine, and you then feed that back to the school.
There are summer options in London but they tend to be five days solid so a bit like jumping in the deep end. See how she gets on with a weekly lesson first and if she loves it, then you have a next step.
Hopefully someone will come along with names of specific places closer to you. All the best.