According to my copy of Martin Middlebrook's The Falklands War :
The plaque was later removed by a Spanish naval officer and taken to Buenos Aires.
Since he says 'Spanish' and not 'Argentinian' I presumed the plaque was removed sometime before 1811 when the Spanish settlements in the Malvinas were withdrawn. Argentina became independent in 1816 and the next settlement was established by Argentinians in 1826, under British protest.
So I did a little digging on Google books and in The Falkland Islands, Mary Cawkell says that the plaque was removed in 1775 by a party from Puerto Soledad. In turn Buenos Aires was captured by the British in June 1806^* and the plaque was taken and returned to London. After that I don't know.
^* Buenos Aires was recaptured by Argentinian forces a month and a half later.