Your challenge is to have the population of Ireland restored to its heights before the Great Famine by the year 1900.
Without waving away the Famine, that's just not going to happen, maybe just maybe if you were able to utterly change the UK's politics to allow massive government intervention (which is ASB honestly) you could slow/arrest the continued migration between the Famine and 1900 but regrowing the population...
Not a chance.
 
I don't think Ireland could have maintained its peak level of population even without a great famine, some form of mass emigration was due to happen.
 
I am reminded that the British government apparently considered resettling millions of Hong Kongers in Northern Ireland in the prelude to their handing the city over to the PRC:


Now, this is obviously well after the stipulations of the OP, but it does cause me to wonder whether there might be some other demographic that the 19th century British Empire might consider resettling on the island.
 
My timeline concentrates will be concentrating on something that mitigate totally the effect of the famine, but does reduce it, meaning that Ireland retains more of its population in a time when it also receives limited autonomy.

Worth checking out, hasn't got there yet but will soon!
 
Maybe a "back home" campaign in the US either as a cultural movement amongst Irish Americans themselves, or more darkly through coercion on the part of some sort of hyper nationalistic regime. But very unlikely before 1900.
 
You would need large scale industrialisation in Ireland and large industrial cities in Ireland.
Hard to see that without ASB
 
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