I see that William & Anna married in 1860:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1860/09589/5477358.pdf
Birth registration only started in Ireland in 1864 so any children born between 1860 and 1863 won’t be in the statutory birth records, but could be in church baptism records. The only child I can see born to that couple was John in 1866 when the family were in Connor (Co. Antrim).
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1860/09589/5477358.pdf
Have you any idea of when and where Anne was born? If so, you could check the parish baptism records. If in Connor, then from 1878 onwards that was the newly created parish of Braid. Its records start that year. Prior to that Connor would have been in the parish of Drummaul whose RC records start in 1825. Newtownhamilton is in the parish of Creggan Lower. Their records start in 1845.
Anne senior’s death in 1890:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
William died in Lurgan in 1896. His son John was the informant and was living in London at that date:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1897/05878/4666929.pdf
I know nothing about French records but there should be experts on Rootschat who may be able to assist. I suggest you post a query on the “Other Countries” Board asking for advice on how to find Anne’s death certificate in Beziers, France in 1893.
https://www.rootschat.com/forum/other-countries/
