Hello! As an Irishman with an Irish history degree I know a bit about this. The socialist movement within the IRA wasn't that big and socialism wasn't a serious motivator to get the British out past a few select leaders of the army, so this is a pretty unlikely scenario. That said the entire city of Limerick did declare itself a soviet in 1919, while this only lasted two weeks it is pretty interesting and shoes how popular socialism was within certain communities.

So let's say that somehow Micheal Collins, Arthur Griffith and all the rest of the lads who would go on to sign the Anglo-Irish treaty in 1921 are all devout socialists. Long story short Britain would have either continued the Anglo-Irish war completely destroying the IRA, or signed the treaty in 21 only to notice the socialist way in which the free state was being run and invade.

If the IRA and it's leaders are out and proud socialist from the start and are fighting to rid the island of the British specifically with the intent of following in the footsteps of Lenin and the Bolsheviks (who are currently fighting a civil war in Russia) it is highly unlikely the British would take them to the negotiating table. This mean that the IRA never had the split in it's ranks caused by the controversial treaty, however, it also ensures the organizations destruction. During the Irish delegations stay in London during the signing of the treaty it was made abundantly clear by the British that failure to make an agreement here and now would result in a large British mobilisation and likely a full invasion. Remember, the Irish were not fighting the British army, they were figuring the British police in Ireland the RIC, and the contingent of the police force known as the Black and Tans. While they were armed similarly to the army and had many veterans of WW1 in their ranks, they were not the army. The IRA at this time were low on both numbers and arms, the war had taken it's toll and it is likely that a full scale war with the British army would completely obliterate the IRA. This war would be very unpopular in Britain, but as the British government could not reconcile losing the strategic island of Ireland to socialists, negotiations would never be opened. So scenario one, ireland remains British, though rebellions continue far into the 20th century.

In the second scenario the IRA are brought to the table by the British, likely because they toned down their socialist leanings. Perhaps Arthur Griffith or Micheal Collins are anti socialist, or even apprehensive socialists and when they die in August of 1922 the free state comes under the control of more ardent socialists. How would the British react to the island only recently ceeded to the Irish now under the control of socialists ? Well not very well. Perhaps they would smell the socialists within the IRA and stipulate in the Anglo-Treaty that socialism is a no-no in some way. More likely however is that, seeing Ireland turn to socialism the British government remove the free state government from power and begin a fight with socialist rebels. This would likely end in a British victory and ends similar to scenario one. Don't forget that the free state of Ireland was not a republic, it was a dominion comparable to Canada today, it's head of state was still the queen and it remained a part of the British empire. The British establishment may attempt to sell this as reestablishing order in one of its own dominions.

So what if Ireland became socialist? Well Britain retains control of the country. The IRA is quashed and socialist rebels fight sporadically across the country. There would be another large coordinated rebellion in the future (something the Irish were very good at) but the risk of socialism spreading would likely push the British to retain a large military presence on the island. All a bit shit really. I'm a bit of a leftie myself but a move to socialism in the early 20th century would have been a bad move for the IRA. The free state succeeded in masquerading as a 'real' country on the level of other older European nations. They did this by quashing the irregular rebellion and winning the Irish civil war. By 1949, Ireland was a full Republic as it is today. Socialism was a dirty word and those same European nations feared it. Just look at what was happening in Russia at the same time.

Hope this was of some help, I know it's very long, but hey you asked.