There's no good indication that separate parts of the world have separate outreach centres to service children who live far away. Each of the main schools that we know of (Hogwarts, Dumrstrang, etc.) seem to be boarding schools, enabling children within their respective catchments to travel to school at their convenience, remain during term time and then return home or stay on site over the holidays if no family are available to look after them.
In the event of a wizard child born to a family from the Falklands, they'd presumably just travel to the nearest port of entry to the mainland UK then travel to Hogwarts via the Hogwarts Express like every other child in their year.
For the record, since the population of the Falklands recently voted overwhelmingly (by a margin of 99.8%) to remain a sovereign British Overseas Territory, it's quite unlikely that any of the families would send their children to the South American school Castelobruxo as their 'home school' despite its proximity to the islands.