David,
According to the 1901 census, Mary Ann was born in Co Down, so I’d suggest that should be the focus of your search.
Tradition was to marry in the bride’s church or denomination. The couple married in St Anne’s in 1882. That was Church of Ireland. Both the marriage certificate and the 1901 census point to a birth before 1864 which is when birth registration started. So you won’t find a birth certificate for her. You may find her baptism. That looks likely to have been in a Church of Ireland church. Not all their records have survived and of those that have, not all are on-line. Those that do still exist are in PRONI, and some are on the rootsireland site (subscription).
Mary Ann’s father was Hugh Nelson, and I read his occupation as “farmer.” I searched deaths in Co Down 1864 – 1901 for farmers named Hugh Nelson, of an age to be her father. I found these 2:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1883/06379/4831169.pdf
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1875/020607/7230535.pdf
You could search the COI baptism records for the parishes that those men lived in.
There are about 7 Hugh Nelsons in Co Down in Griffiths Valuation (c 1860). You could go through them and determine which was a farmer from the type of property they had, and then search those church records (the parishes are all shown in Griffiths).
If Mary Anne’s father died pre 1864, the search will be a bit harder but given that she was only born a year or two earlier, you have to hope he was still alive then.