This is a very interesting scenario which raises a lot of questions and possibilities. There is a chapter in a book on alternate/counterfactual history in which something like this scenario is envisaged.
Probably Ulster (defined as the six counties of OTL) would be excluded from Home Rule for a number of years, so a bicameral Parliament would be established in Dublin for 26 counties. The Home Rule Act provided that the executive body would be the Executive Committee of the Privy Council of Ireland, so John Redmond would most likely be the equivalent of Prime Minister (though that term was not in the Act), under the authority of the Lord Lieutenant. Here is the relevant wikipedia entry:
http://en.wikipedia.org/wiki/Home_Rule_Act_1914 .
Redmond would have supported Britain in the Great War and organised a recruiting campaign in Ireland as he did in OTL. He would have been opposed by Sinn Fein, but they would have had no more support than they did in OTL until the executions after the Easter Rising, which does not happen in this scenario.
The British government's attempt to extend conscription to Ireland in the Spring of 1918 is successfully resisted, though it is widely supported in Ulster. By now I expect John Dillon would be the chief minister in Ireland following the death of Redmond on 6 March 1918.
I assume that the outcome of the Great War is the same as in OTL, with a general election in the UK on 14 December 1918. Under the terms of the Home Rule Act 42 MPs from Ireland would be elected to the Westminster Parliament. In the postwar election I expect in the region of 10 or 11 Unionists to be elected, with the others some variety of Irish Nationalist, possibly including a few Sinn Fein members.
At the same time there would be an election to the Irish House of Commons which would likely result in a majority for the moderate Nationalists led by John Dillon.
After the Great War there would be growing demands for more powers to be given to the Irish Parliament. I don't know how the Lloyd George Liberal/Conservative coalition would react to these demands. There would be no Irish War of Independence as there was in OTL.
Questions: How popular would Sinn Fein become? Would British political parties organise and contest elections in Ireland?
How long would Ireland, outside Ulster, remain part of the United Kingdom?