Well, don't think the UK would deploy nuclear weapons that easily. Maybe after huge loss of life, or at least prospects of huge loss of life. No, they would maybe have invaded Argentina, with such a coastline and losses in the war, British would have invaded a lot of the country to force a surrender of the islands. The US never deployed nuclear weapons in their cold war "regional conflicts", not did the Soviet, I think compared to these parties the UK was more rational.
I tend to agree it was a lost war, or at least a very short sighted one. There was a tiny chance of the UK having difficult deploying troops such a long way from home, and would have given up the islands to avoid conflict... but considering the UK's government, the Argentine government made a bad judgement call. I think it didn't help Argentina to get the islands back at all, without the war, maybe something would have been settled by now, but now the UK has fought for it so recently... not likely. But I think Argentina has more pressing concerns now than a couple of Islands.
Now let's go back to the facts, instead of this hypothetical/political talk.
About the Vulcans, they were put into action largely as a deterrent, in the same way Lybia and Iraq did with their Tu-22s. Making a statement. Conventional bombing, although not very effective with these bombers, sends the message that they are in theatre and mean business.