Given Stephen Lyons was born in Cork in 1827 and emigrated to Bermondsey around 1851, I am going to put some background here on the situation in Cork at that time. Of 1596 entries in Griffiths for the surname "Lyons", 20% are in Cork and Cork City, 13% in Galway, 9% in Limerick, 7% in Kerry with the other 51% of the Lyons mentions in Griffiths being scattered across all the other counties. Cork has the most Lyons in Ireland
Finding Stephen Lyons in Cork covers the search in more detail. An original feeling was that the most probable ancestor is Stephen Lyons, born to Patrick Lyons and Mary Daly at Ardprior, Buttevant Parish on 3 February 1827. Another possibility was Stephen Lyons born to Edmund Lyons and Mary Warren on 23 February 1829 at Ardskea, Buttevant Parish. But, certainly by 2014, and earlier, these possibilities have been ruled out, as 1. my Stephen called his firstborn Timothy, and I am almost certain his parents were Timothy and Honorah-the latter dying in Bermondsey 1870 & 2. due to DNA evidence .
The Irish National Archives has a list of genealogy websites to investigate
The two maps below show the spread of Lyons geographically in Griffiths around 1850. Each circle shows the number of Lyons families per Civil Parish. The second map, lower down, confines positioning to only showing mentions of "Stephen Lyons" and "Timothy Lyons"
Care has to be taken to distinguish between "civil parishes" "catholic parishes" and "church of Ireland parishes". Further the Roman Catholic church has an odd division of Dioceses with a finger of Kerry Dioceses coming into the area that we are most interested in. I feel that I have undertaken a thorough research of the records in the National Library of Ireland . It is probable that we are not going to make much further progress unless we can find a Lyons locally in the Mallow area (I have written to all in the Phone Book) or from a family researcher on the web with additional records.
. In 2014 an all 37 DNA marker match was found between my cousin, John Lyons- son of Len, and a descendant, Paul Lane, of a Stephen Lehane ~1800 and of a Denis Lehane 1829. & in 2015 an all 37 DNA marker match was found between my cousin, John Lyons- son of Len, and a descendant, of Daniel Lehane 1843 , So I am now looking into Kilnamartyra and Ballyvourney where these Lehanes seem to have come from.: areas 20, 53 & 7 on the maps above
More historical background to the Lyons family