There seems to have been two separate Cole Lane's - one was off Henry Street in the north city centre, the other was in the south of the city near Meath Street. The first would be in St. Mary's Civil parish, the 2nd probably in St. Catherine's.
You didn't mention a denomination for this Carney family - if they were CofI then any parish records may include occupations which can help match up records.
I had another search of records on the IrishGenealogy website, but not seeing any likely matches Williams baptism in Dublin city.... There are a number of RC baptism for the name William Carney (with variations) around the right time, i.e. between 1800 and 1820, but none that I can see with father Nicholas.
Some
possible links to siblings... i.e. baptisms with father's name Nicolas Kearney in Dublin City between 1800 and 1820 :
1. Alexr. Kearney in 1801 to parents Nicolai and Eleona. No residence is recorded, and the parish for this record is St. Andrews RC - which covers parts of the east of the city near Trinity College, and stretching west to Grafton st. etc
See :
Alexr. Kearney - 18012. In 1804 there's an RC baptism with in St. Michan's RC parish for a Judith Kearney with parents Nicolas (as Nicola) Kearney and Mary Byrne.
See :
Baptism of Juditha Kearney - 1804 3. In 1809 there's yet another baptism in a different parish with father's name Nicholas Kearney.
See :
Laurentius Kearney - 18094. And a John Kearny to Nicholas and Elizabeth of 29 Marlborough St in 1818.
See :
Baptism of John Kearny - 1818 see previous posts :
Re: Street names Dublin Shane