It predates Britain's joining the EC. By WW2, Canada and New Zealand already insisted that they were independent countries with their own foreign policies, and declared war on Germany separately (however, the conservative government in Australia insisted that as a dominion, it was bound by Britain's declaration of war).
The issue of self-government for the dominions (and for Ireland) was recurrently debated in Britain in the second half of the 19c. Some people proposed intercontinental federalism, and one made the argument that steamships could travel from London to Melbourne in the same time early-18c sailing ships took to travel from London to the northern end of Scotland. Pre-devolution Britain was incredibly centralized, so the only real alternatives for the dominions were federalism and (gradual) independence, and at the end federalism went nowhere, so they became independent, by war in Ireland and slowly and peacefully elsewhere.
The basic problem is that, without splitting England, there was no way to institute any federal structure once the US was gone. In 1901, England had 30 million people, the rest of the then-UK 9 million, Canada 5.5 million, Australia 4 million, New Zealand 1 million, and the Cape Colony 2.5 million. You can't have meaningful federalism when one unit has nearly three-fifths of the total population of the federation - you'd get a very unequal situation, like Prussia's dominance over the German Empire and Weimar Republic. The intercontinental nature of such federation would compound the inequality, since federal decisions would be made in London, weeks away from Canada and Australia by steamship rather than Berlin's location hours away from southern Germany by train.
The same problem crops up today, leading to the unstable combination of devolution to the non-England units and Tory calls for English votes on English laws. Nowadays there's a meaningful force, Labour, attempting to split up England into regions, but a hundred years ago, there was no such force.