Claire,I may have come up with something.Using Irish naming patterns as a guide,I found 2 births for John Delaney,one of which I believe could be the fellow you seek.
John Delaney father:Charles Delaney born 1825 co Laois
John Delaney father:Lawrence Delaney born 1828 co Laois
In Griffiths Valuation 1848-1864 there are the following Lawrences and Charles in county Laois:
for Lawrence: townland:Ballynagallparish:Killabbin townland:Balinlaparish:Kyle townland:Glenconraparish:Offerlane townland:Clashparish:Offerlane townland:Clonterryparish:Ardea townland:Kilmainhamparish:Ardea townland:Rosskeltonparish:Clonenagh and Clonagheen
for Charles: townland:Boleybegparish:Dysartgallen townland:Laught/Commonsparish:Castlebrack
What I would do first is to find the marriage cert of John and Anne.It probably took place between 1850-1855 in New York going by the age of their oldest child Mary born @ 1855.The marriage would probably give you the name of one of John's parents and it would determine which baptism record is his.Was John naturalized? It might be worth looking into.Also try to get his and Ann's death certs as well.Exhaust all sources in the US before going to Irish records.If I can help more,let me know.Kim