Possibly best to start with Australian records (so much available in newspaper, etc.) before trying to find the family in Ireland especially since it's not yet clear exactly when this branch moved to Australia.
Here’s an outline of what’s posted on one online tree-
https://www.wikitree.com/wiki/Algeo-50 Robert H. Algeo (1787 Leterkenny, Donegal-1872 Manorhamilton) m. Anne Butterfield
[note: SantaT says Robert Newcomen Algeo]
1. William Butterfield Algeo (1816 ‘Co Lutrim’- aft.1866 NSW) m.(1856 Manning River NSW) Mary Ann Lockhart
a. Charles Lockhart Algeo (2 June 1857 Manning River NSW-26 Feb.1943 Bundaberg Qld) m. Fanny Crossingham- family
b. William Henry Algeo
c. Norman Lockhart Algeo
Another Algeo topic (Australia)-
https://www.rootschat.com/forum/index.php?topic=565391.0Charles Lockhart Algeo married Fanny Crossingham Charles Lockhart Algeo was born on 2 Jun 1857 at Scotts Creek, Manning River, the son of William Butterfield Algeo, a surgeon, and his second wife, Mary Logan (nee Lockhart). William and Mary were married on 5 Aug 1856. Mary Logan had come from Edinburghshire, Scotland, with her daughter, Sarah, on the ship ttBeett. William Butterfield Algeo was born in County Leitrim, Ireland, the son of Robert N Algeo, Lt Colonel, in the Portuguese service, and Anne Butterheld. Harriet, the first wife of William Butterfield Algeo, died on 28 Feb 1847 aged28 years.
http://static1.1.sqspcdn.com/static/f/822236/22016148/1361767596563/crossingham.pdf?token=avn76Jxo5fFifdwuumsTpDU4oy0%3D