Ian Robertson has asked me to respond to your query on his behalf.
I’ve spent most of today on this one and am not getting very far.
There are several variants of CORCORAN, e.g. COCKRANE and COCKRAN. There are also CORCORANs by default due to the slip between pronunciation and pen, e.g. COCKERELL, COCKERILL and perhaps your COCKERY as well as CORCKERY.
This would make the earliest arrival in Australia, barring the navigational expeditions, the following. However, he was neither a convict nor Irish.
Without limiting information, all I can do at this stage is to give you those most likely Tasmanian connections. Going beyond Tasmanian shores involves literally hundreds of families and possibilities, especially amongst the CORCORANs proper. I don’t know what to suggest as a likely search strategy, except exhausting all possibilities. I have not included any Tasmanian convicts simply because those with the relevant names would be too late [1825 is the earliest], if, as you say, W. W. CORKERY's father was one of the earliest of the CORKERY/CORCORAN transported to Australia.
Surname: COCKERILL Given Names: William Sex: M Event: Death Registration Place: Hobart Event Date: 24 Aug Event Year: 1839 Age: 79 Registration Year: 1839 Registration Number: 222 Rgd Number: 35 Source: Tasmanian Pioneer Index, 1803-1899, Deaths
This William had a relationship with a Mary PARROTT which produced one offspring, George. George seems to have moved to the Victorian goldfields and married a Mary VANCE. None of their children was called William.
Surname: COCKRANE Given Names: William Sex: M Event: Death Father: William Registration Place: Hobart Event Date: 01 Nov Event Year: 1853 Age: 2 Registration Year: 1853 Registration Number: 586 Rgd Number: 35 Source: Tasmanian Pioneer Index, 1803-1899, Deaths
Surname: COCKRANE Given Names: Lillian Maud Sex: F Event: Birth Father: William Mother: Clara BOYCE Registration Place: Portland Event Date: 21 Apr Event Year: 1888 Registration Year: 1888 Registration Number: 2819 Rgd Number: 33 Source: Tasmanian Pioneer Index, 1803-1899, Births