My guess after Peron dies and before 1982, Peron was known to have Nazi sympathies and would have been to controversial, so the 1960's or 70's.
Actually, there were negotiations during Peron's third term, including joint sovereignity/codominium
(link in Spanish)
http://www.lanacion.com.ar/1455991-la-propuesta-secreta-de-los-ingleses-a-peron-por-las-malvinas
The British non official paper to the Argentine government
https://es.scribd.com/doc/83966769/...cret_password=1j9fyl7n1et8o1qpetyh#from_embed
Reportedly, Peron agreed enthusiastically because, while the terms weren't the absolute (and impossible) best, he said "Once we put a foot there, they won't ever kick us out". But he died some three weeks later and the imbecile of third wife and vicepresident torpedoed those negotiations. Have a POD in which Peron chooses a competent VP and the codominium agreement goes through. If that butterflies away the 1982 war, then you have an entirely different political and demographic situation in the islands within a few decades. Changing "codominium" to "Argentine sovereignty" isn't out of the question then.
Anyway, for an outright purchase, the end of WWII can be a good opportunity. It can be painted in the same light as the nationalization of the railways, so the public can buy it. Or, in a more dramatic note:
For whatever reasons, only a few British ships can follow the Graff Spee in its way to the South Atlantic in 1939. With better odds, the Graff Spee manages to defeat her pursuers but, battle damaged, limps to the Falkland Islands for emergency repairs. Fog of War/lack of proper communications (uh, short wave radio was already a thing back then. Maybe the German sailors confiscate those?) make the world believe a German invasion of the Southern Atlantic is happening (yeah, only one ship, but it's not like the Falklands were heavily inhabited) and panic ensues. The RN, for whatever reasons (early Norway? Just normal war issues?), can't quickly dispatch a force to retake the islands.
In Buenos Aires, the pro-British OTL governments buys into the panic. In OTL, after the sinking of the Graff Spee, the Argentine government considered taking an official pro-allied-yet-still-neutral stance. Have the media print covers with "The German Navy has invaded Argentine soil", and butterflies might cause the Argentine government to dispatch the Argentine navy to the Falklands to fight off the Germans
as an aid to the United Kingdom. Argentina becomes a formal member of the Allies and, among other things, garrisons the Falklands, since the RN is occupied elsewhere (ie, all over the world). When WWII ends, transferring sovereignty to the one non-commonwealth ally who declared war on Germany all the way back to 1939 and had been "taking care" of the islands for almost six years isn't far-fetched.