The ability to launch a nuclear device is not the same as the intention to use it in the present conflict. Russia moved 2 SSBM's to south-osseatia in the Georgian war recently, but did anyone really think they would use those to nuke Georgia? Nuclear weapons are political weapons, not military weapons, especially if we talk about city-buster size weapons.
First things first, if we want a seriously escalated falklands war we need to establish a POD that for one, makes the UK take the situation seriously and for argentina to actually be a serious military power, something akin to a strong military buildup by Argentina during the 70's plus a very strong support for the president instead of the way it went historically. Combined with a resurgence of nationalist "Europeans/Yanks out of S-America". Then a strong show of support for argentina by its neigbours. But that would require an almost entirely different global situation and a POD at least 10-15 years before. Then possibly with considerably larger naval engagements that go very badly for the UK, then bringing in Nato and such then yeah, we might see UK using the bomb during the Falklands war. But as it was in historic, the probability of UK using the bomb during Falklands war is about the same as UK using the bomb against quebec, so far up the realm of ASB that it might actually make an interesting storyline.
If UK actually asked all its historic allies for committment + did a real heavy duty diplomatic offensive + all things needed and prepared to level the military junta ala Bush and Iraq they could. Anyway, UK using nukes needs something about 10 times more dangerous to UK than the falklands war, only realistic situation i could even remotely think of for that post cold war that cant be solved with conventional weapons, diplomacy or allies would be something like i described earlier.