1. At the outset, I assume that your mother is not an EU citizen and this is what is the problem. Additionally, I assume that when you refer to "we" you are referring to yourself and your husband having Italian citizenship. In such a situation, there are two options open to you. The first is because you are an Italian citizen, along with your husband, you can seek EU Treaty rights under Directive 2004/38/EC, which would allow yourself and your husband get a five year visa for the UK, to include your dependent mother under the terms of this Directive. In the UK, the directive is transposed into the Immigration (European Economic Area) Regulations 2006 amended by SI 2009/1117 and amended by SI 2011/1247. Applications cost £55 and an additional £55 per dependent. The Directive extends to all family members, including a dependent parent. This would allow your mother to stay with you in the UK.
2. The second option is different, in that it would involve living in the EU outside of the UK and Ireland. There is a Directive on Family ReUnification - Directive 2003/86/EC - which would allow your mother to live with you in any other EU country such as Italy. This Directive has not been adopted by the UK or Ireland but applies in all other EU countries and could apply to you because you are Italian citizens. Your mother would come within the terms of this Directive and it would allow you to live in any other EU country.
My mother has an Italian passport as well
My grandfather was born in Italy
3. If your mother has an Italian passport, then she has the right to live and work in the UK. Under the EU freedom of movement provisions, any EU citizen, such as an Italian passport holder can live in the UK. So there is no problem, even if she has dementia.
What should I do if they try and send us back to South Africa?
4. They cannot send you back to South Africa. YOu need to show both your own and your mother's Italian passport to the authorities you are dealing with. The authorities are obviously labouring under the impression that you are South Africans.
5. A court order can always prevent any deportation.