The US won't allow that. Because a Nuclear Latin America will be an end to US dominance in that region.Countries like the UK should get tough instead of the soft state image they are promoting these days.
Heck they don't even have a Carrier and there were even reports about the reduction of their Nuke arsenal in the future.
How is the first bold possible without the second bold? Their expeditionary capability is almost halved because of having no carriers. Their budget cuts are going into billions, there are no civilian jobs, forget military spending.
OTOH, Latin America is rising and becoming regionally more united. This is not 80s where West European countries could just walk over any former colonies. Believe me if UK was to consider a military approach, this time it would be tough as Brazil and Venezuela are also involved. Brazil might have strong relations with USA but in recent political decisions, it has shown that it doesn't have to toe US lines all the time (Iran sanction vote, enrichment issue etc). Brazil is quite a strong nation.
And we all know how lovely Chavez is. He has a small country but a cash rich (oil) and a rapidly arming country. Sanctions from US only made him double his armaments from Russia. He's ploughing into the Russian defense goods and that is pretty strong for a SA country.
Apart from this, Argentina is now more vehement in its position. Closely cooperating with other regional countries of South America, Argentina has a strong say in the continent and with that Fernandez woman in power, it is only set to get stronger.
US won't do anything. Period. What benefit does supporting UK, serve US? Nothing. At the maximum, it will mediate and ask for hostilities to end. But at this stage when Europe is dead except for France and Germany, if UK attacks Falklands they would badly lose.
Nuclear threat would be the worst idea as Nagraj said. Remember that after Iran and North Korea, we don't need any more nuclear states. Since South America doesn't have one, there will be voices raised for them to get nukes using UK's hypothetical nuke threat as an excuse. And leaders like Chavez would be only happy to fund such a program.
This means USA would have additional headache in the region and that is something it doesn't want. So in the end, if the hostilities heat up, this might cause frictions between USA and UK instead more than US and South America.