Before 1932, the Dominions were not independent nations. In theory, the British Parliament could still pass legislation for them. In practice, though, they hadn't done so in awhile as the legislatures of the dominions took care of all their own internal business. When the Statute of Westminster was passed in 1932, the British Parliament lost this power and the Dominions became independent nations in theory as well as in fact, united only by the fact that they all had the same monarch as Great Britain.
(Note: It's important to remember that the monarch, while the same person, was technically not the monarch in a Dominion as he or she was in Great Britain. Australia is not ruled by the Queen of the United Kingdom, but by the Queen of Australia. They just happen to be the same person.)
You can see the change in the relationship in how the Dominions reacted to the outbreak of the First World War vis-a-vis the Second World War. In 1914, it was pretty much, "Well, Britain is at war, so that means Canada, Australia, New Zealand and South Africa are at war, too!" No questions asked. In 1939, however, each Dominion debated for itself and then declared war on Germany on its own.
Up until the 1980s, some Australian and New Zealand legal cases could still be appealed to the Judicial Committee of the Privy Council in London, the British Parliament still had the right to pass legislation for the Australian states (though not the Australian federal government) and London had to give its consent to certain changes to the Canadian Constitution. These ties were severed by the 1982 Canada Act and the 1986 Australia Act. Amazingly, some New Zealand legal cases could still be appealed to London as late as 2003!
In some Commonwealth Realms, the Judicial Committee of the Privy Council still functions as a sort of "Supreme Court" to this day. When I was vacationing in the Bahamas awhile back, there was some sort of major land dispute that was causing an acrimonious legal battle, and the main headline in the local paper was that someone was threatening to "appeal the case to London".