There is absolutely no plausible scenario in which Britain uses nuclear weapons against Argentina or anyone else in relation to the Falklands Islands.
It should be remembered that Britain spent most of the 1960's and 1970's trying to negotiate to get rid of the Falklands. The entire premise of the war was not for Britain to regain a piece of territory that important, or even to liberate occupied Britons, but about saving face on the world stage and showing that Britain - a world power - would not be pushed around by a second rate south American dictatorship.
Therefore if Hermes is lost, with or without Invincible, and the US "offer" of borrowing one of their carriers (manned or unmanned) is not forthcoming, then Britain withdraws her fleet for the winter, but maintains submarines and blockades the islands. In the meantime a new task force is properly planned and prepared and sails in the summer, probably around October time.
As it was the closest the Argentine airforce came to sinking one of the carriers in the war was the Exocet strike on Atlantic Conveyor. This represented the last Argentine Exocet attack (as they ran out) and conceivably the last time they were ever likely to be able to take out a carrier, short of managing to get one of their subs beyond the destroyer/frigate screen, which also never happened.
The far more likely scenarios for British defeat in the war, are Argentine air force bombs being modified so that the fuses arm quicker and actually detonate, rather than failing to go off as happened so many times. The British would have lost many more destroyers and frigates this way, and might eventually have been compelled to withdraw after making their landing. The other way is to actually attack the British landing ground with a coordinated strike between the Army and airforce whilst they are still trying to unload, the point at which the entire British invasion was most at risk. The Argentine Army however lacked the leadership, motivation and importantly coordination ability with the other services.
The unelected manderines of the Foreign office might have tried to shift the Falklands but elected ones were "not about to hand over 1500 British people to a bunch of Fuc$ing nazis" - so your first statement is not accurate - however the effect of their (the civil servants of the Foreign office)manouverings did play a part in making the Argentine leadership (coupled with their own ignorance - for example they claimed to a US Diplomat that Britain had not fought in a war since 1945!!!!) beleive that Britain would not and could not go to war with them over the Falkland's.
The issues with the Argentine armed forces having delayed fused bombs being dropped too low or submarines not being able to penetrate the carriers defensive screens comes down to a lack of any real operational experiance outside of a very one sided war against and insergency plus a couple of Coups compared to a Military that had lots and lots of real operational experiance and very good training.
But yes having another 6 warships damaged by bombs might very well have that effect. Or at least delayed the landings while more ships arrived to replace them.
I did once read that rocket Pod attacks where actually very effective at 'mission killing' British warships but this type of method was very rarely used
However once 3rd Commando Brigade was ashore with its artillery, a bridgehead established and Goose green secured with the airstrip adapted for Harrier GR3s - it was pretty much game over. The Argentine commanders should have realised this and sued for terms.
By the time of the Atlantic Conveyor attack - Britain had the ability to jam the Exocets (Thanks in part to the French who supplied much data on the weapon to the British) - it was just unlucky that the last 2 missiles were fired at the Civilian ship and not a military one as either carrier would very likely have had counter measures, jamming and in extremis the design of both carriers and their highly trained all volunteer crews to allow them to survive such hits.
I think its highly unlikely that a pair of exocet missiles with their 165 kg warheads would damage a 17,000 or 23,000 warship to such an extent that they would be sunk.
Mission killed possibly - but sunk no.
Worst case scenario - as you say - Falklands war 2.0 recommences 6 months later