When it comes to the oil companies, I doubt that the nationality of the company will be important, nor will who it really matter who has ownership of the land.If Oil is found in the exclusion zone Stanley will turn into Aberdeen, with Anglo-Australian, UK, US & Canadian companies flocking to the islands thus expanding the current population, this represents a worst case scenario for the Argentines because it ties in more than just the UK. The US needs to secure future reserves and is wary of South American politicking, having a friendly source of carbons in the South Atlantic is a win, win scenario for them.
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All the nations you just mentioned are Anglo and Protestant orientated, and they will be digging for oil in an area that is mostly Catholic and Latin, this will cause more anger across Latin America. This is going to get interesting if a massive deposit of oil is found, and Latin nations see themselves as getting poorer and the Anglos richer. This would play right into the hands of leaders like Chavez and friends.
I think you'll find the oil companies regardless of where they are based are businesses - in real terms they will just want stability, a reasonable assurance that all their plant, equipment and output from the region won't be 'nationalised overnight. At this stage, that probably looks like the UK, however if the Argentines (or even the Calathumpians) do get the islands through negotiation or force, then be under no misapprehension, BP, Shell, Exxon, Gazprom etc will be looking to make a profit from all that oil, regardless who's flag is flying from government hose in Stanley.