No, just no. Thatcher was not a madwoman and, even for someone with her refreshingly 19th century views, employing nuclear weapons (even as a demonstration) would be breaking the clear 35-year precedent that no major powers would employ them except in a MAD scenario. Plus, the US was being kept informed on British plans, and had considerable leverage with Thatcher. She would not go nuke without letting the US know. There is simply no way the Reagan administration would allow Britain to use nukes. If the British campaign or invasion went south, I suspect that Reagan would be the one who acted to ensure the Falklands were returned to Britain, not with nukes, but with strong-arm diplomacy followed if necessary by real aircraft carriers, B-52s, and US marines if need be. And the Soviets would do nothing.