Thank you for your question and patience, I’m Tom and I’ll try to help you.
She is no automatically eligible for UK citizenship because mothers were not permitted to pass on their citizenship historically in the same way that fathers were. So unless your mother already registered here as a UK citizen then she would not yet be a UK citizen.
This means that she would have to apply for registration if she is eligible.
You have correctly identified the Form by which she would apply to be registered. This is the form avialble to children of mothers who were not able to pass on their citizenship and the guide to making such an application is contained here:
The documentation commonly required is contained at page 12 of the above guidance. If you have your mother’s previous two expired passports
then you will have to send this. Obviously you will also need your sister’s birth certificate showing that your mother was her parent. Obviously, if your mother’s name as listed on your sister’s birth certificate is different to her passport then you will have to show the applicable marriage certificate for your mother to demonstrate that your mother is the same person as named on the birth certificate.
If your sister currently only has leave to remain until the 6th December then the application will not be decided until this time. I note that you have not said that you have got your sister’s birth certificate and presume that you will not be able to get it in time, so although I appreciate that she may not want to return to SA it would probably be in her best future interests to do so. The reason is because if her application fails for some reason then she will be an overstayer which would mean likely mean that her future applications for visas to come her would likely be refused.
My goal is to provide you with a good service. If you feel you have received anything less, please reply back as I am happy to address follow-up issues specifically relating to your question.