If you are looking for John Irvine’s birth in Cookstown c 1818 then you need church records. Statutory birth registration didn’t start in Ireland till 1864 and before that we have to rely on church records, where they exist. To do that you need to know the person’s denomination. You have mentioned that James married in a Catholic church in Scotland but do you know if he was originally RC himself? I ask because, looking at the 1901 census for Tyrone, there are 191 people named Irvine, and all are either Presbyterian or Church of Ireland. Not a single Catholic.
Cookstown is in the parish of Derryloran. The Church of Ireland records start in 1795 and have been transcribed. They are on-line on this link. I had a look around 1818 but do not see James. You might want to double check as well as look for the other names you have listed:
There are 3 Presbyterian churches in the parish. 1st Presbyterian’s records start in 1836, 2nd Presbyterian in 1822 and 3rd in 1835. The RC records start in 1827.
The RC parish records are on-line free on the National Library site:
The Presbyterian records are not on-line. They have been copied and are held in PRONI (the public record office) in Belfast. A personal visit is required to view them. Obviously they don’t cover the years when John & James were born but you might find Andrew & Hugh’s births.
If the family was Presbyterian or Church of Ireland then that tends to point to them being descendants of families who settled in the Cookstown area as part of the Plantation of Ireland in the 1600s. They likely came from Scotland.
Here’s Pigots Directory for 1824 detailing what Cookstown was like then:
There are 2 watch and clockmakers listed, named Samuel Brown and Robert Gillmore. Perhaps your ancestor was apprenticed to one of them?
You ask about connections between Cookstown and Lanarkshire. Huge numbers of people left the counties of Ulster for Scotland all through the 1800s. Ireland lacks natural resources. No coal, oil, iron ore etc, and so apart from a modest amount of shipbuilding in Belfast and the Belfast linen mills (which mostly only employed women), it did not really get the industrial revolution that benefited England and Scotland where mills, steelworks, ship building, coal mining and all their support industries were major employers creating hundreds of thousands of new jobs. Much better paid than subsistence farming or weaving. Added to that you had the effects of a massive population explosion in Ireland – up from 3 million in 1750 to 8 million in 1841 (no-one is really sure of the reasons why but reduced neo-natal deaths seem to be a factor) and the famine. So some push factors and some pull factors saw huge numbers of people leave Ireland. Something like 8 million people emigrated from Ireland between 1801 & 1921.
If you look at the Scottish censuses for the Glasgow area in the late 1800s, you will see that about every fifth person recorded there was born in Ireland. Scotland was a particularly popular place to go to work because it was easy and very cheap to get to. Several sailings every day from Belfast. The shipping companies main business was cargo and the passengers were just top-up revenue. Competition was fierce and passenger fares very low. People working in Scotland could come home for weddings or the harvest, as well as holidays (Glasgow used to shut down for 2 weeks every July, known as the Glasgow Fair holiday and there would then be a huge exodus to Ireland). You could also send children back to stay with their grandparents, thereby leaving the wife free to work. You couldn’t do all those things so easily from Australia, America or Canada.
Lanarkshire was full of coal mines, steel works and other heavy industries. There were thousands of new jobs going there all through the 1800s. Going from Ireland to Scotland (or England) and then on somewhere else eg the USA was very common. It’s called stepped migration.
James Irvine appears on a tree on Ancestry (Gatt family). It notes that Andrew b 1831 and John b 1820 were James’ half brothers. It isn’t clear what the source is for that information. No parents names are recorded.