The nurses? Do you really think a nurse has the expertise, knowledge and qualifications to diagnose aspergers? I believe that a diagnose for your mum doesn’t change anything. She will be the same before or after that.
Social services can get involved for emotional abuse (and many other kinds of abuse, It doesn’t need to be physical). If your circumstances have changed since then, give them a call. They will intervene if they think there is a cause for worry, but it will be probably for your sister as you are, I assume, less than a year from being, officially an adult, so even if you were in care you might be out of their remit by now.
Considering your A level results will shape the options you have for the future, I agree that you need to concentrate in your studies now. This is not a time to be thinking of moving out or sorting your mum, when you are, I assume, very close to sitting your exams. You can think about other decisions afterwards.
If you don’t have enough time/space to study at home, go to the library, a cafe, a friend’s home and create that space. If you are not at home you don’t need to worry about constant distractions. The key is, again, to concentrate in your A levels, they are the passport to your freedom.
You will not manage to change your mum or help your sister in such a short time, concentrate in what has more impact in your future life (again, your grades). I would say you don’t even need to worry about finding a way to move out as you will be doing that without major hassle or recourse to support agencies by becoming an undergraduate student.
Best of luck.