Hi Richard,
As Shane advised there is a lack of records. Have you looked at all his records in New Zealand as some countries record the names of parents on their death certificates. Also look for obituaries as his life story may have been recorded.
Do also look at the wills available at https://genealogy.nationalarchives.ie and the land records on Griffiths Valuation, though making links with the names is difficult. Also look for records for headstones on Billiongraves, findagrave, historicgraves etc. There are more and more volunteers photographing and transcribing same. Concentrate on the names he gave his children in the hope that he named some of his children after his father. When you look for deaths look to see was he in attendance when any of his relations died, it is a long shot, Alwell is not as common as Clarke or O'Reilly so their occurrance is less frequent.
Also contact the following, Wicklow in the chance that they have access to church records which were not fully transcribed and Cavan in case they have records not yet transcribed. It would be normal that if someone got married outside their parish that notification would be sent to their home parish. Sometimes the genealogical offices have queries from other people that may be looking for the same people as you.
Genealogy Centre Contact Details
Wicklow Family History Centre
County Archives, County Buildings, Station Road, Wicklow Town, Co. Wicklow.
Tel: +353 404 20126
Email: wfh@eircom.net
Genealogy Centre Contact Details
Irish Family History Foundation
Cana House
Farnham Street
Cavan
Co. Cavan
Email: enquiries@rootsireland.ie
In looking for Alwell, do consider other spelling variations such as Olwill etc. But in the hope that someone in Co. Cavan recognises this family you could write to The Editor of The Anglo Celt through linda@anglocelt.ie you may reach a wider local audience. There is no guarantee they will publish your story but if you have photos it would help the story along.
Regards Carmel.