If your DD gets a diagnosis, you will be told something like: 'She meets the threshold'. They won't tell you a 'scale'. A long time ago I believe they did tell people where they sat on diagnostic scale but it was decided it wasn't actually helpful.
I think it's to stop the idea that having 'high functioning' autism is in some ways not 'really' a serious barrier to learning/social interaction. This is also the reason that the term Asperger's isn't used any more.
I'm not sure if it will be any different with a private diagnosis but when my DS was diagnosed, we were given the news and pretty much sent on our way.
We did get a 20-page report though which has proved very useful as we used it as evidence when we applied for an EHCP for him. There were also some suggestions in there for the school in terms of supporting him.
There may be something called Autism Outreach in your area. Once a child has a diagnosis then someone from that team can come in and see DD at school and speak to their teacher.
As you're going private though, if you do get a diagnosis, you may have to chase this up yourself. Or speak to the school Senco and they will do it.
There may also be courses that the council runs to support parents. I didn't do the course offered to me. Some of them are better than others but I do hear it's a good way to meet other parents and talk through your feelings/swap tips etc.
If you do get a diagnosis, even if you're expecting it, it can come as a shock. It can take a while to process. My DH cried.
I think of my son's diagnosis as a 'sign post' which helps both him - and those supporting him - to put things in place so he can navigate the world better.