Statutory recording of Protestant and nonconformist marriages started in Ireland in April 1845. I can’t see the marriage of Joseph and Jane listed in the indexes. That, and the children’s ages (eg Wallace born c 1846) suggests they married before that. In which case you will be reliant on church records. To do that you’ll need to know their exact denomination and the parish they lived in. Otherwise it’s a needle in haystack. Few of the parish records are on line. If you do narrow the search and find where they came from, copies of most parish records are in PRONI, Belfast. Same problem for the children’s births. Birth registration only started in 1864 so you won’t find any birth certs for children born before that.
I searched Scotlandspeople for the death of Jane Nelson (ms Johnson). There were no matches for that information. Nearest was a death for a Janet Nelson previously Johnston who died in Calton, Glasgow in 1886, aged 69 (ie b c 1817.) You might want to pay to see the cert to see if it’s the right one. (Scottish death certs give spouses names as well as the deceased’s parents, so can be very helpful).
If you find Joseph’s death in Scotland, it should give you his parents names. It would then be worth checking the surviving part of the 1831 census for Co Derry to see if his father is listed. (Joseph himself was probably too young in 1831 to be a head of household which is all the 1831 recorded by way of names).