The process of independence in "settler countries" was much different than in countries that had been colonised in the true sense. You are right to highlight the difference.

Let's start with the ideological basis for British colonisation. Officially, Imperial rule was a kind of tutelage; "the coloured peoples of the world" were being improved and brought to civilisation, and they would be obviously worthy of independence after being uplifted by the white man. Unofficially, of course, it was about extracting as much economic value out of the colony as possible. But raw greed wasn't really something the British wanted to tell themselves. So under their "white man's burden" type rubric, things like self-government were clearly not possible; white people would "have" to do that instead until some nebulous future where the local people were somehow deemed "worthy" of that.

The settler countries were already white. They held British citizenship (in AU and NZ until 1980), held the right to free movement throughout the Empire (until at least the 1960s, and in some places the 1980s) and understood themselves as culturally and racially British. They were inside the privileged elite, sharing in the ideological superiority that implied.

Practically, they were also far away from the centre of Imperial government in London. They were given self-government, almost immediately. New Zealand had a parliament with universal property-holding male suffrage, including Maori, in 1857. The Australian states all had their own parliaments as well.

Think of Empire as being like the EU framework in this instance, not like a single country. Although "officially" the Empire was in charge of all defence and foreign relations, in reality the individual places had a broad range of things they could do independently. When friction occurred over the possible inadequacy of the defence of Australia, for instance, the Australian states simply federated in 1901 and combined their strength. They won the right to their own navy in 1908. An independent foreign policy was also possible; the US Great White Fleet came to Australia and New Zealand because they had been invited by the local governments, over Home Office objections. The Australians also restricted immigration outside of Empire, again over UK objections, with their White Australia policy.

Most every step towards "independence" for these settler countries as a politically acknowledged reality were pushed by the British themselves. The first of these was probably the Imperial War Council formed in 1917 - although the Settler countries entered the First World War as a bloc, they controlled what they sent. The South Africans, for instance, didn't send anything near per capita what the other Dominions sent. And these places had a proprietary interest in their young men, and how their resources were being spent - hence the council. Although the settler countries therefore entered the war with the UK, they each signed the Treaty ending the war individually, and argued for their own positions at the table. (On an unhappy note, the first thing Australia did was thwart a clause asserting racial equality.)

So it was clear that each Dominion was conducting its own internal affairs pretty much as it liked; the Westminster Parliament, however, was still technically the supreme lawmaking body in Empire. This was clearly dumb to everyone. After two Imperial conferences to talk things over, in 1931 the Statute of Westminster was proposed;

"No Act of Parliament of the United Kingdom passed after the commencement of this Act shall extend, or be deemed to extend, to a Dominion as part of the law of that Dominion, unless it is expressly declared in that Act that that Dominion has requested, and consented to, the enactment thereof."

This needed to be individually ratified by each parliament to become internal law. This is still not independence; the Settlers are still British citizens, with British passports, who think of themselves as British. They understood themselves as Australians and New Zealanders and Canadians as well, but those things were also inherently British. It was probably for those reasons (and the sound security and economic reasons to be IN Empire) that meant this statute was actually adopted very slowly. Canada de facto accepted it right from the outset, but it took until [edited] the fall of Singapore in the Second World War for it to be ratified in Australia. New Zealand only ratified it in 1947, solely in order to move towards abolishing its Upper House.

This process was fairly satisfactory until it was clear Empire gave the Settler colonies nothing. Britain retreated militarily after 1960, declaring its intention to not extend its power beyond the Suez. It also joined the European common market, cutting the trade links that had made Empire so valuable to the Settler countries. This process was very painful - not just a retrenchment of power, but also of identity. It was a process of coming unmoored and of a sudden searching for new identities. This was the time the Settler countries did things like get new national anthems and create new founding legends, but also politically of the Constitution acts. The 1980s saw the Canada Act of 1982, and the Australia Act and the New Zealand Constitution Act of 1986. (By this stage the South Africans had been a republic for quite some time.)

So the short answer to your question is; the long slow process of de facto independence, which took a long time to complete.

This is still a highly condensed version of this history, so if I have been unclear at any point, please ask for clarification.