Search for William Neal/Neill born abt. 1818

Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line.

RC records are mostly on-line on the nli site:  

https://www.nli.ie/en/family-history-introduction.aspx

For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:

https://www.nidirect.gov.uk/publications/proni-guide-church-records

If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net

You really need to get William’s parents names from his death or marriage certificate, to be sure of finding the right person. William Neal is a pretty common name. I can see 29 in the 1901 census of Co. Tyrone. It would have been equally common in the early 1800s.

Regarding the spelling of the name, the idea of a single or correct spelling for a surname or a place name is very much a recent phenomenon designed to meet the needs of modern officialdom. Before that, especially in Ireland, there was no consistency. Names were spelled phonetically and each variation was down to the whim of the particular person recording the information. You will often see the spelling change as the records go back. This rarely indicates a deliberate decision to alter the name, nor even a mistake. Not everyone was literate, but even when they were, exact and consistent spelling simply wasn’t something they bothered about. In addition to varying the actual spelling, O’ or Mac prefixes were optional and were often omitted.  So you need to search under all likely variants - Neill, Neil, Neal, O’Neill etc. They are all the same name.

Thank you for your help