The first Nicholson was probably Robert Nicholson who married
Anne Williams at Holyhead on 9 December 1787. He is described
as an innholder in the Quarter sessions for 1793 and a warrant
was issued by the High Sheriff of Anglesey to "take" Nicholson
the same year. The Poole papers at Gwynedd Archives mention
a case Smith v. Nicholson in 1793 which might also be him.
Further snippets of information about the Nicholsons of Holyhead
can be found in the records of the Anglesey Quarter sessions.
I could speculate a little and suggest that his wife may be the Anne
Nicholson who was still alive aged 85-9 in 1841 and his children may be
daughter Anne living with the old lady, son Jared (married to Judith Owen
in 1816) who lived next door, and daughter Frances (married to Philip Walsh
or Welch in 1814). The younger Anne Nicholson had a son John with John
Spenser in 1819 and he presumably took her surname, but Jared's earlier
child out of wedlock with Sidney Rowland in 1814 presumably took the
mothers name.
If I am right these families would account for all the Nicholsons in
Holyhead in the 1851 census.
Further info at:
http://www.a2a.org.uk/ global search for Nicholson Holyhead
http://www.gwynedd.gov.uk/DATRhagorol/rhagorol/index.english.pl search for Nicholson
http://isys.llgc.org.uk/ search for Robert Nicholson
http://freepages.genealogy.rootsweb.com/~hughwallis/IGIBatchNumbers/CountyAnglesey.htm#PageTitle see the marriages & christenings for Holyhead
Bob