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Short but very bloody conflict
In this case, the Irish keep going; they might manage to take a few border towns and then start to get pretty shredded. The US is a nightmare scenario domestically Reagan, providing actual military assets to the British against the Irish is going to hurt a lot, but if the Brits invoke article 5 and he does nothing it makes NATO look like it's not worth the paper it's written on so massively costs him abroad ( unlike Argentina they can't really claim the exception for colonial wars applies which to be honest was a bit of stretch even there ) plus of course the ussr will be milking this for everything it is worth.
My suspicion is the US effectively bribes the Brits to not invoke article 5, offering them diplomatic support to the hilt on both Ireland and Argentina, and a great big chunk of economic aid to help them through this 'turbulent period', while internationally trying to reframe this as not the Republic of Ireland V the UK but instead some sort of rouge (and presumably communist sponsored) cabal taking over a peaceful Western nation, and trying to destroy NATO and the Irish peace process with a pointless war ( does not particular matter whether this is true or not it's just the only narrative they can really do let them save face). In part due to this the US will absolutely want greater regime change more then just a few resignations at the top for this, and will almost certainly get it.
In the war itself the British take some casualties but the Republic's Armed Forces are pretty much annihilated, If the government still reesists and the Brits have to start heading into the Republic proper I suspect things get quite nasty with lots of potential for civilian casualties etc. My prresumption is they basically put together an armoured column and make a beeline for Dublin, sieze the different mechanisms of state and at that point anything short of ASB mind control will force a surrender. This is unlikely to impact the Falklands war at all ( amusingly possibly making their lives easier as the US is more willing to back the UK diplomatically just to avoid the British feeling they have to invoke article 5 over Ireland to reduce pressure on the home front), the bigger challenge the British for the Falklands was getting their forces there, which is greatly unaffected by this and they can handily manage with troops still on the home islands.
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