Irish neutrality was directly linked to De Valera and Fianna Fail who came to power in the 1930s having reluctantly recognized the Irish Free State in the late 1920s entered the Dáil contested elections won thumping victory over WT Cosgrave and the victors of the Irish Civil War a decade before. De Valera was determined to dismantle the Treaty get rid of land loan repayments dating back to the Parnell era buy out of Anglo Irish landlords, get back the Treaty ports to keep Ireland out of future British wars, dismantle British powers over the government of Eire as Ireland was renamed and pave the road for a full constitutional independent Republic.
De Valera did not declare a Republic while he was in office even during the Emergency because he did not want to politically abandon Northern Catholics cut off from the south under Protestant majority sectarian rule in Northern Ireland.
Twenty years on from the start of the War of Independence 1919-1921 there would be many men who have taken up arms if the Irish state Ireland took sides against Nazi Germany. Ultra conservative Catholics who had supported Franco most definitely would have seen this as a betrayal. Die hard nationalists in the ranks of the IRA would have taken the same line. Marxists around the world taking their orders from Stalin did not support WW2 until the Nazi invasion in 1941 and ideologues in socialist republican ranks would have been no different.
De Valera was almost shot after the 1916 Rising and could easily have died during the War of Independence and Civil War like many of his peers. It was not certain that he would emerge as the leader of Fianna Fáil led government in the 1930s after being sidelined by the IRA leadership during the Civil War 1922-23. Catholic Ireland could have adopted fascist rule possibly under Eoin O'Duffy who led the Blueshirts or some other hard line nationalists. Most of Europe adopted militaristic fascist style government at that time. If Michael Collins had lived and had filled the role De Valera had filled it is possible an Ireland less triumphantly Catholic - Collins held strong anti clerical views - and with friendlier relations with Britain - Churchill and Collins apparently grew to like eachother - might have emerged. Collins might have been quite like General Pilsudski of Poland - posing as a patriot soldier and politician.
In this scenario Collins would have allowed Royal Navy ships to use Irish ports allowed British troops and aircraft to use bases in Ireland while British money weapons and equipment would have been used to raise Irish divisions to fight alongside the Western Allies in North Africa Italy France Germany and Asia just as the Canadian South Africans Australians Indians and New Zealanders did. Luftwaffe air attacks on Cork Waterford and Dublin would have been devastating but outraged the Irish people against the Nazis.
American troops and aircraft would have been based across Ireland just as they were in the UK and because many would have been Irish American they would be warmly welcomed by the Irish people. However the "oversexed overpaid and over here" phenomenon that led to a sexual revolution in wartime Britain would also have played out in Ireland.
Irish troops would have landed in Normandy and perhaps an Irish parachute division like the American British and Polish units would have taken part in drops in Holland and Germany.
Post war Ireland would be part of NATO might have joined the EEC sooner enjoyed the benefits of the Marshall plan abandoning economic isolation industrializing during WW2 and benefitting from the post war boom without the plague of emigration. American and British air ground and nuclear forces would have been based here.
In the 1960s more positive relations between Britain and Northern Ireland and Ireland might have existed due to the experience of war. Perhaps more reasonable government without discrimination against Catholic might emerged with bigots sidelined thus avoiding decades of the Troubles?