How is Oceania properly pronounced in British English? Is it /ˌəʊʃɪˈɑːnɪə/, or /ˌəʊʃɪˈɑːnə/? I know a lot of people who use the latter, but I have always been taught the former.
My answer does not directly address the issue of which pronunciation of Oceania is "proper"—an objectively narrow question that moderators Andrew Leach and tchrist address in comments beneath the posted question. Instead, my answer looks into the question of why multiple pronunciations of Oceania may have arisen in the first place. If that aspect of "Pronunciation of 'Oceania' in British English" doesn't interest you, I urge you not to read the rest of this answer.
It may be relevant to popular pronunciation of Oceania that the word Oceana has coexisted with the word Oceania in English writing for many years. Here is an Ngram chart comparing the frequency of occurrence of Oceania (blue line)" versus Oceana (red line) for the period 1802–2008:
As you can see the two words appeared in books in the Google Books database with roughly equal frequency until about 100 years ago—and Oceana has by no means dropped off the face of the earth since then, although Oceania has certainly become considerably more common.
One early source of Oceana is James Harrington, The Commonwealth of Oceana (1656)—a presentation of a theoretical commonwealth based on England under Cromwell. The Commonwealth of Oceana is a classic of English political thought and continues to be studied (and reprinted, at least as recently as 2015).
James Froude, Oceana, or England and Her Colonies (1885) is noteworthy for its focus on Australia and New Zealand—the heart of the region that we now call Oceania—which Froude seems to see as a realization of the superior commonwealth envisaged by Harrington.
One might wonder then whether Oceana was not the original name for the area at the southwestern edge of the Pacific Ocean and the southeastern edge of the Indian Ocean. That appears not to be the case, however. Matches for Oceania appear as early as 1819, and with some regularity beginning in the 1830s. From Abraham Rees, The Cyclopædia: Or, Universal Dictionary of Arts, Sciences, and Literature (1819):
ADRASTÆA, so named by professor De Candolle, from Adrastea or Adrastis, a surname of the goddess Nemesis, who was the daughter of Oceanus ; because the plant in question is a native of New Holland, which has been called by some persons Oceania.
From "[Statistics of the World]," in Gray & Bowen, The American Almanac and Repository of Useful Knowledge for the Year 1830, taken from Adrien Balbi, Balance Politique du Globe (1828):
The surface of the earth has been estimated at 148,522,000 square miles, of 60 to the equatorial degree (geographical miles), of which nearly three-fourths, or 110,489,000 square miles are covered by the Ocean and the interior Seas;–the remainder, consisting of 37,673,000 square miles, forming the five parts of the world, called Europe, Asia, Africa, America, and Australasia (or Oceania).
It isn't clear whether the parenthetical "(or Oceania)" reflects wording in Balbi's original chart or is an interpolation added for English-speaking audiences who were at least as familiar with Oceania as with Australasia.
From "Oceania, Malaysia, Australia and Polynesia," in Niles' Weekly Register (October 9, 1836):
OCEANIA. Under this comprehensive head are included all those numerous islands, groups of islands, and all that great island continent, New Holland, which is spread over the Pacific ocean, between the two continents of Asia and America. As to mere extent on the sphere, these Oceania regions from Sumatra to Easter island inclusive, with the equator very nearly as a middle line of latitude, extends through above one hundred and fifty degrees of longitude. An area so vast demanded subdivision, and it has been divided into three great sections [namely, Malaysia, Australia, and Polynesia].
Other frequent matches for Oceana over the past two centuries refer to Oceana County, Michigan, and to a law book publisher doing business as Oceana. An early (1834) match for Oceania refers to a genus of hydroids (medusa jellyfish). More recently, of course, Oceania appears as the name of the superstate where Winston Smith lives in Nineteen Eighty-Four (1948). I would not be at all surprised if Orwell had named his dystopian commonwealth in a bitter nod to the optimistic use of Oceana in Harrington (and maybe also in Froude).
It is difficult to gauge the effect of the works of Harrington and Froude on how people pronounce the South Pacific region now commonly termed Oceania. The references to Oceana in those works may have had no bearing at all on popular pronunciation of Oceania, or they may have contributed to a situation in which speakers have long had to pick their way through mixed instances of Oceana and Oceania without, perhaps, being fully aware that both spellings were in competing use and logically invited different pronunciations.