How about if Argentina invades in 2003, during or immediately after the invasion of Iraq? By then, British forces were too stretched to respond immediately. The Royal Navy had a significant force to support the Iraq War, including the carrier HMS Ark Royal. The British Army is stretched, having deployed large forces in both Iraq and Afghanistan, and the Royal Air Force has significant resources deployed there to support them.
Also, the British Army has a fairly large contingent deployed in Germany, and a significant RAF contingent on Cyprus.
So British forces are stretched, and mobilizing a proper response will be a long and hard process. The US can invoke the NATO charter if the British attempt to withdraw from Afghanistan, and push hard for Britain to keep forces in Iraq, while pressuring Argentina and leading a sanctions drive. At the same time, it will probably be impossible to take the islands without the Royal Navy's ships deployed in the Persian Gulf, especially Ark Royal. If Argentina does not evacuate, or if insufficient pressure is applied, the British will have to choose between disengagement from Iraq or letting the islands stay permanently Argentine.
I suspect that the British would begin thinning down their RN contingent in the Gulf at the very least, and would probably take at least some army and RAF forces out of the country and maybe from Afghanistan and Germany. Putting them together will take time, time enough for Argentina to properly prepare.
I would expect Argentina, if it is smart, to have learned from its past mistakes and stage a massive military buildup. This means upgrading its air force and navy, and maybe buying anti-ship missiles. The Argentines will be wise to spend the time fortifying the islands, because a British counterattack, like last time, will always be on the verge of disaster. If the Argentines create one disaster like sinking a carrier, its over.