The CVR was recovered even before the civil air investigation comission personel arrived (it took them about 1.5h to get to the site). Police at the scene handed it over to them. Later the same afternoon, the civil air accident investigators found the FDR.
Still, it took a couple of days to have them sent to England to get the data retrieved. Because they weren't in that good of a shape, it was deemed that the UK air accident facilities were the ones with the highest probability to successfully recover them.
The ($%&·) argentinian news channel just made the whole thing up about having the transcript of the last words of the pilot.