There had been a similar situation under the previous Labour government, who had dealt with it very quietly by send one of our nuclear subs down there and letting the Argentine govt find out.
Under Thatcher, the Foreign office were aware of a similar threat and did warn of it. It was ignored by the cabinet. Max Hastings wrote in his book on the Falklands War an account of how Richard Luce, a junior F.O. minister was so concerned he stood up in the Commons and spoke about it and was absolutely eviscerated, not by the opposition, but by his own backbenchers.
Instead, the signal we sent was to continue with John Knott's defence cut's, which were much greater than he wanted, but forced on him by the Chancellor and PM, and withdraw even the Ice Patrol Ship Endurance.
The Argentine military Junta, desperate to deflect attention from the home economy and out of control inflation, concluded that the UK no longer cared enough about the Falklands and lacked the capacity and the will to recover them.
Ironically, Luce and Carrington fell on their sword and resigned after the conflict ended, on the basis that the failure to avoid a war is a failure of diplomacy and their office, when actually they were the two Tories not culpable. Even more ironically is it led to Thatcher, who was most culpable and whose days had been numbered due to her disastrous economic policies, unemployment and inflation, being re-elected as a hero, rather than a villain, with a stonking majority, in part due to an ineffective peace loving opposition leader, to effectively wage an economic civil war on significant sectors of her own populace (miners strike) with a militarised and politicised police force, the fall out and dislike from which continues to this day. Not to mention the squandering of North Sea Oil revenues, the dismantling of British manufacturing, setting the path for de-regulation of the financial sector. Austerity (and therefore Brexit and the capacity and readyness of the NHS to deal with covid-19) has it's roots in the 1980's and sheer incompetence leading to an avoidable war.