Dear Clare:
Thank you for your post to the Ireland Reaching Out message board.
I note that the McCarthy surname is not a common one for County Tyrone. While there are McCarthy baptisms/births registered in County Tyrone under the Roots Ireland database, there isn't any reference to an Andrew. However, not all of the baptism registers are on the Roots Ireland database and it might mean that his baptism record is in one of those that is not on Roots Ireland.
However, I did take a look at the 1901 Census, and there are 3 McCarthys noted in that Census. The purpose of this was to try to narrow down a few possible locations in County Tyrone for your ancestor. The link for you to view is below:
http://www.census.nationalarchives.ie/search/results.jsp?searchMoreVisible=false&census_year=1901&surname=McCarthy&firstname=&county19011911=Tyrone&county1821=&county1831=&county1841=&county1851=&barony=&parish=&ward=&townland=&houseNumber=&familyId=&ded=&age=&sex=&search=Search&ageInMonths=&relationToHead=&religion=&education=&occupation=&marriageStatus=&yearsMarried=&birthplace=&nativeCountry=&language=&deafdumb=&causeOfDeath=&yearOfDeath=&familiesNumber=&malesNumber=&femalesNumber=&maleServNumber=&femaleServNumber=&estChurchNumber=&romanCatNumber=&presbNumber=&protNumber=&marriageYears=&childrenBorn=&childrenLiving=
Sorry for the long-winded link!! However, this might be a good starting point and then you can work backwards. There are registers on the National Library of Ireland website for County Tyrone at the link below:
The link for Omagh is Drumragh parish. Here is the link:
https://registers.nli.ie/parishes/0365
The link for Cookstown is Desertcreight parish. Here is the link:
https://registers.nli.ie/parishes/0204
Again the area listed for the third McCarthy falls under the Omagh area (Clanabogan DED and Aghadulla townland). That would be the first link. The fact that there are 2 families in this parish with the surname McCarthy might be a further clue as to the origins of Andrew McCarthy.
I did note the surname of McCarty under the Roots Ireland which you have referred to. There isn't any entry for a McCarty in the 1901 Census.
As to the search for Margaret Toner, there are a number of entries for her--none for 1864, but one for 1865. You might include her parents' names if you know them to see if they are also listed on the 1901 Census.
For ease of refernce, the 1901 & 1911 Census records are available at this link:
http://www.census.nationalarchives.ie/search/
If you need any further assistance, please let me know. If you are able to narrow down the parish, please reply on this thread and we can try to arrange for a volunteer to meet you.
Kind regards,
Jane.