There are two things to comment on. First, although others have tiptoed around this, no one has come out and said it clearly: It's a matter of who is doing the colonizing and who is being colonized. European colonialism was profoundly disruptive everywhere (even in Europe but that's a different question), and it had more or less disastrous effects on indigenous populations. These range from virtual extermination (many Native American groups, though clearly not all), to expropriation (most Native Americans, Australian aborigines, New Zealand Maori) to exploitation (Africans, South Asians).

One reason that Australia, Canada, and the United States are doing well is that the dominant white populations of those countries were and remain the colonizers. Policies of land settlement, legal structures, and economic development all favored the white populations of these places at the expense of non-white populations, be they indigenous inhabitants of landscapes killed or forced off their land, or enslaved populations. As such, while the process of colonialism was obviously disruptive for everyone involved, those disruptions were explicitly geared to benefit some people at the expense of others. Australia, Canada, New Zealand, and the United States (as well as South Africa, arguably, and other countries if we wanted to dig deeper) are the continuing legacy of that process.

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The second point I'll comment on deals with the economics of empires after the turn of the nineteenth century. European colonialism combined with the industrial revolution to reorient the global economy in profound ways, what we call the "Great Divergence" (from Pomeranz 2001, but now a common term). Europe and the British settler colonies became industrialized "core" areas, while most of the rest of the world became primary-producing "periphery," a relationship that led to much greater rise in per capita income in the core areas (Argentina would be an interesting comparison and Japan obviously presents an exceptional case). Findlay and O'Rourke argue--synthesizing a great deal of literature in the process--that we can essentially think of colonies and post-colonies in the 19th century in several groups: the British offshoots, whose success is conspicuous; former sugar colonies, which almost universally suffered from global competition in sugar production outside the mercantilist system; non-sugar colonies and former colonies which suffered a great deal of political instability, in particular Latin America, though why this was the case is a bigger question; non-sugar colonies or former colonies that remained relatively stable politically but did not industrialize, like India; and "new" colonies, including most of Africa.

The category that doesn't fit here are the British settler colonies, the ones you asked about in your original question. Aside from Europe and the British settler colonies, the "rest of the world" frequently came to rely on exports of primary products to the industrialized regions of the world, and found itself forced into a new and dynamic global economy in subordinate positions. Places like India and much of Africa were exposed to global markets and global competition, but were generally not afforded anything like the tariff protection that Europe and its British offshoots enjoyed. Of course, Latin America also erected substantial tariff barriers in the 19th century, so that doesn't explain the divergence on its own. We do know, however, that the British settler colonies preferred to trade with Britain and that Britain preferred to trade with them (speaking the same language reduces transactions costs broadly), and thus the US, Canada, New Zealand, and Australia all emerged in interesting roles in the global economy: they alone provided substantial primary products to the European "core" and Britain in particular (cotton, wool, grain, meat, timber), and they developed their own industrial base, substantially financed by British capital.

If we push our analysis a bit and look beyond the purely political relationships between colonies and metropoles, it becomes obvious that Britain, the US, Canada, Australia, New Zealand (and perhaps South Africa) can be seen as a global capitalist empire: landscapes and resources around the world were subordinated to the benefit of their white inhabitants, financed by British capital.

See Findlay and O'Rourke, Power and Plenty (Princeton, 2007)

This question could be easily re-phrased to "Why are some nations that have experiences colonization considered western nations or developed nations and others are not".

I think the fundamental difference between India, African and Central/ South American colonies and Australia, Canada and the USA is culture and ethnicity. In Australia, for example, we considered ourselves as a natural extension of Britain up until the post-war era because we culturally considered ourselves as British. There was no real process of decolonisation in Australia because we never felt the need to resist Commonwealth influence. Instead we slowly gained more independence as we drifted towards the United States, politically, economically and militarily. Today we are a self-determining sovereign nation with friendly relations with the Commonwealth of which we remain a part. Also, although Australia, Canada and the United States all had indigenous populations upon colonisation, the invaders quickly oppressed and subjugated them so that they posed no threat. The ramifications of this are still being felt today in all three countries as there still exists a significant social inequalities between the indigenous populations of those countries and the rest of society. Politically, the populations of Australia, Canada and the United States was dominated by white people which also helped to put them on an equal footing with Europeans.

Conversely, the developing nations that have experienced violence and unrest since decolonisation have had an entirely different experience because European influence has essentially divided their societies and cultures in one way or another. The countries you named as examples of this also had rich established cultures and societies preceding colonization. This naturally led to resistance and division which ultimately led to decolonization. These nations have had a long history of exploitation and western interference which has continued to fuel the fire.

Whilst this is a very brief overview which only touches on the subject I will recommend some further readings:

Phillip McMichael. 2008. Development and Social Change: a global perspective. 4th edition.

Stephen, E, Ambrose. & Douglas, G, Brinkley. (1997). Rise to Globalism. 8th edition.

I think the best predictor of a country's governance post-colonialism would be its governance pre-colonialism. The "power vacuum" you discuss is a function of the extent to which colonies were administrated as actual countries, with bureaucracies and rule of law, as opposed to extractive colonies.

Acemoglu, Johnson, and Robinson write about the different legacies of extractive colonies, like in the Belgian Congo, and "Neo-Europes" like in Botswana. They argue that the extent to which Europeans governed and created political institutions determined the level of governance left behind after colonialism. Extractive colonies lacked institutions and fell into the corrupt, neo-patrimonial states we see today in places like DRC or Sierra Leone.

With this argument in mind, it is important to realize that French Colonialism was different from British Colonialism was different from Belgian Colonialism etc. The Portuguese for example were arguably the worst, building no state institutions and leaving behind a mess in Angola and Mozambique, two countries that struggled with civil war for a long time (Cold War politics didn't help either, as the United States and Soviet Union effectively used southern Africa as a proxy war). The British were arguably much better at creating state institutions, particularly in places British people went--New Zealand, Australia, but also in places like Ghana.

Which brings up another important point--not all British colonies were governed the same or saw the same outcome. Daniels, Trebilcock, and Carson argue that the extent to which indigenous people were included in political administration and the extent to which local political systems were integrated into colonial administration predicts the ability of post-colonial states to self govern. For example, Tswana political integration, like the use of local chiefs and granting greater autonomy, helped make Botswana a much stabler, more developed country than Kenya for example.

If you look at Canada (and Australia), british colonists effectively wiped-out any meaningful existing aboriginal society (and in the case of Canada, severely curtailed the french society).

But again, in Canada and Australia, if you look at it from the aboriginal viewpoint, you are experiencing thost "terrible consequences" daily.

In Africa and India, however, colonists never really wiped the existing societies there, so they linger on with all the damage colonial rule.

Isn't the answer simply that when the colonizers killed off the indigenous people, or otherwise marginalized them to the point of being nearly insignificant minorities, the eventual country did well (for the people who weren't dead, aka the ethnic descendants of the colonizers) (e.g. USA, Canada, Australia). When the colonizers simply ruled over the indigenous population without exterminating them, when they left, there remained a huge group of indigenous people with most of their government/culture/society long having been removed from the social memory. Surprisingly, these people usually don't fair very well.

One other major consideration is disease. In Canada, the US and the Caribbean, disease played a major role in allowing European expansion. The spread of diseases, like small pox, was done unintentionally and intentionally and caused the deaths of untold millions of North American indigenous peoples. Which in turn made it increasingly easy for expansion. Today there is virtually no indigenous populations left on any of the Caribbean islands.

The opposite holds true for expansion into Africa, or South Asia. Indigenous peoples of Africa had already been exposed to the European diseases that caused such havoc in the new world. And Europeans soon found that they were very susceptible to the diseases of the tropics, like malaria.

Brazil is an interesting case of these two scenarios intersecting in one country. As the Portuguese settled the coast and southern (more temperate) Brazil the native peoples quickly fell victim to disease and died, or were killed, off. Yet that vast Amazon rainforest was not settled because it was perceived, right fully so, too dangerous because of disease. Because of Brazil's close proximity to Africa they could purchase African slaves for much cheaper than say, Americans. Because of that the Brazilian slave system developed into one that relied on Africans, not African Brazilians whom had been born into the system of slavery. And almost immediately these Africans began fleeing into the Amazon rainforest and forming quilombos. They survived for various reasons but one reason was that these were people that came from the tropics and sub-tropics of Africa where malaria and other diseases were endemic. Therefor they were better suited to deal with the diseases of the tropical Amazon rain forest than any of the Europeans.