That's not quite how their Wikipedia entry puts it:
On 31 August 1903 the Chagos Archipelago was administratively separated from the Seychelles and attached to Mauritius.[17]
In November 1965, the UK purchased the entire Chagos Archipelago from the then self-governing colony of Mauritius for £3 million to create the British Indian Ocean Territory (BIOT),[18] with the intent of ultimately closing the plantations to provide the British territory from which the United States would conduct its military activities in the region. On 30 December 1966, the United States and the United Kingdom executed an agreement through an Exchange of Notes which permit the United States Armed Forces to use any island of the BIOT for defence purposes for 50 years, until December 2016,[18] followed by a 20-year optional extension (to 2036) to which both parties must agree by December 2014. As of 2010, only the atoll of Diego Garcia has been transformed into a military facility.
In 1967 the British Government bought the entire assets and real property of the Seychellois Chagos Agalega Company,[19] which owned all the islands of the BIOT,[20] for £660,000[21] and administered them as a government enterprise while awaiting US funding of its proposed facilities, with an interim objective of paying for the administrative expenses of the new territory. However, the plantations, both under their previous private ownership and under government administration, proved consistently unprofitable due to the introduction of new oils and lubricants in the international marketplace, and the establishment of vast coconut plantations in the East Indies and the Philippines.
Between 1967 and 1973, the entire population was removed against its will from the islands and moved to Mauritius and the Seychelles to make way for a joint United States-United Kingdom military base on Diego Garcia.[22] In March 1971, Seabees, United States naval construction battalions, arrived on Diego Garcia to begin the construction of the Communications Station and an airfield. To satisfy the terms of an agreement between the United Kingdom and the United States for an uninhabited island, the plantation on Diego Garcia was closed in October of that year.[23]
The Chagos Islands and their inhabitants were bought and sold like commodities. They appear to have had no choice in the matter.
I rather think that had the entire population of, say, the Isle of Wight been removed against their will to make way for an American military base British people would not be relaxed about it...