Ann,
This may be Emily’s birth in 1874 in Downpatrick:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1874/03165/2160944.pdf
Here’s Emily & Nicholas in the 1901 census of Belfast:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Dock_Ward/Trafalgar_Street/1003994/
Here’s Nicholas’s birth in 1896. Father was a sea captain:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1896/02186/1827172.pdf
The birth certificates indicate the parents were married but I can’t see a marriage in Irish records. Perhaps they married elsewhere. I note Emily said she was a widow in April 1901.
Deaths here of 3 other of Emily’s children:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1897/05850/4657669.pdf
Twins here:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1899/05807/4643726.pdf
(The birth & death certificates indicate Evangalo wasn’t the twins’ father).
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1899/02054/1785783.pdf
There are numerous trees on Ancestry which give this family. Evangelo is reported to have drowned when his ship the Ithaca sank in the Channel in April 1900. Depending on his nationality and the ship’s nationality, his death may be registered in the Maritime register in GRO Southport, England. Or it may be registered in his native country which I assume was not the UK or Ireland.
Son James Alexander King, to father James King, born June 1900:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1900/01988/1765181.pdf
The birth certificate indicates the parents were married but I don’t see any marriage in Irish records.
I looked for Nicholas in the 1911 census, in the workhouse but I don’t see him listed, nor do I see his mother. You mention Nicholas was put in Lisburn workhouse. If the family were in Belfast then he would likely be put into that workhouse. But whichever it was, both Lisburn & Belfast workhouse admission records still exist for 1910/11. They are in PRONI (personal visit required to look them up).