Quinie,
Birth registration didn’t start in Ireland till 1864 so if Mary was born in 1861 you won’t get a birth certificate for her. You might find her baptism but to do that you’d need to know her precise denomination and where she lived.
I searched the marriage records for Alexander Stewart in Co Londonderry 1845 – 1861. There were about 10 but I only found one married to a Nancy. That was in 1852 in Magherafelt. She was Nancy Wallace and they married in the Church of Ireland. (Tradition was to marry in the bride’s church). RC marriages weren’t included till 1864 so if you think the couple married in that denomination then the marriage won’t be in the stautory records.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1852/09434/5417882.pdf
Ideally you need to get Mary’s mother’s maiden name from a suitable source eg her death certificate in South Africa to ensure we identify the right marriage.
I can see Mary Stewart’s marriage to Robert Skillen, registered in Whitehaven, Cumberland, Jul – Sept 1878 Volume 10b page 773. You can order a research copy of that from IPS Southport.
If the couple who married in 1852 are the right couple then I can see 3 children born post 1864 in Magherafelt (for pre 1864 births you need church records):
James 28.11.1864
Ann 3.2.1868
William 6.6.1869
Here’s James:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1864/03599/2327424.pdf
The forenames Anne & Nancy are interchangeable in Ireland so you often see it change. Sally/Sarah and many others are similarly interchangeable.
If you think this might be the right Alexander & Nancy, I’ll see what else I can find. The family doesn’t currently appear to be on any tree on Ancestry.