I believe it was governed as a single unit by the British, and I believe the islands all gained independence roughly around the same time.
The dialects, peoples, customs, and cultures are very similar, certainly not any more different than the differences across the United States, for example. There are natural benefits that derive from federation that are easy to appreciate, and there are natural weaknesses from being tiny and isolated that are easy to appreciate. Newfoundland and Canada, for example, did not win independence at the same time but saw that union was beneficial.
So why isn't there a single country called BWI? Indonesia and the Philippines are many scattered islands, and they are united. So the fact of being scattered islands can't be the simple reason.
Some in fact are still united in tiny groupings: Trinidad and Tobago, or Antigua and Barbuda, etc.
They did. It was called the West Indies Federation and it existed from 1958 to 1962. It included Jamica, Trinidad & Tobago, Barbados and all the Leeward and Windward islands. After it fell apart the islands became colonies again and many would only gain full independance during the 1970s and 1980s.
The Federation fell apart for a couple of reasons. First it was imposed by the British wanting to better organize these small colonies, as a result it didn't have the same kind of 'buy-in' and legitimacy as other federations in the empire that were bottom up efforts. It was the leaders of the Canadian colonies that met first, sorted out the federal structure then asked London to approve it. Because the British imposed the federal structure local leaders did not feel invested in maintaining it. The lack of emotional investment by leaders and populations meant it was easy to put provincial needs and desires first at the expense of national goals.
The second, and most important reason why the federation fell apart was that it was very decentralized and lacking in financial resources. The lack of budget in particular meant the federal goverment lacked a carrot to keep provinces in line and create national policies. Further custom and freedom of movement barriers still existed causing conflict.
As a aside, to respond to part of your question, the 'natural benefits of a federation' are actually pretty rare to apperciate and are very abstract. People like local control of things and are very wary of trusting some person far away to run things that directly impact their lives. [It took two referendums and alot of dealing to get Newfoundland to join Canada] Running a federation is very very hard and requires a constant dialogue between the federal hub and the provincial forces.
With regards to the premise of the question, it's worth mentioning that Canada did not emerge as a single country, but came together in a piecemeal fashion over the course of 150-odd years. OP acknowledges that Newfoundland came along later, but that was only the most recent acquisition:
1791: establishment of Upper and Lower Canada
1841: unification of Upper and Lower Canada into the United Province of Canada
1867: unification of several colonies into the Dominion of Canada
1870: acquires the North West Territory and Rupert's Land
1871: unification with the Pacific colonies
1873: unification with Prince Edward Island
1880: acquires the British Arctic Territories
1949: unification with Newfoundland
By contrast, the British West Indies only took their first crack at unification in 1958. It failed, but then again, Canada's first attempts were not terribly successful, either - armed rebellions racked the country in the 1830s, and the successful unifications some 30-40 years later were in part a reaction to this violence. It's not clear that the natural benefits of federation are really that easy to appreciate at the time ;-)
I don't have an answer to the question but I think it ought to be mentioned that some parts of the British West Indies never became independent and have the status of British Overseas Territories.