Steven Den Beste has some observations on the role of the British King or Queen in Canada and Australia. Most of the things he says are right, however there are one or two issues outstanding, so here goes.
I believe that de jure the British Parliament actually did some power over Canada before the Canadians actually wrote and ratified a constitution about 30 years ago. Until that point, the actual charter for government in Canada was a grant of power from the British Parliament called the "British North America Act". But de facto the British Parliament had none and I don't recall hearing of any attempt by the British Parliament to try to exercise any such influence in the latter half of the 20th Century. (Not that it would have done them any good to try.)
As one of Den Beste's Canadian readers points out at the bottom of the post, the British Parliament's de jure power was removed in 1931, when the British parliament passed a law called the Statute of Westminster. In this act, the British parliament specifically gave up the right to legislate for "the dominions" of the British empire (Australia, New Zealand, Canada, South Africa, Newfoundland, and Ireland (which was not yet a republic)), unless the parliaments of these countries specifically asked for it.
The "specifically asked for it" clause was necessary because in some instances the dominions did not have the full legal structure of independent nations. In particular, Canada's constitution was the British North America Act, which could only be amended by another act of Westminster. Therefore, if the Canadians ever wanted to change it, the British had to be involved. This "involvement" would contain no actual influence over the legislation, but would just consisted of passing whatever bill they were asked to pass. Australia did not have this particular problem, because our constitution contained a provision for amendment via referendum, and we could therefore sort out any legal difficulties without involving the British.
However, it is still more complicated than this. In Australia, the Statute of Westminster was viewed with some ambivalence, as Australia believed that if it became fully independent the British might be less willing to defend Australia in a war with Japan. For this reason, the Statute of Westminster was written in such a way that it did not apply to Australia until Australia passed another act of its own parliament officially adopting the statute. This did not happen until 1942, when Britain had failed to successfully protect Australia from Japan and Australia wanted to be absolutely certain that its troops were only subject to Australian law.
Plus, the Statute of Westminster only applied to the federal government in Australia (although in Canada it applied to all levels of government) . In theory, even after its adoption the British parliament could legislate to overrule Australian state law, but not federal law. This was not cleaned up until the Australia Act of 1986, which was an act passed by the British parliament on the request of the federal and state parliaments in Australia, the purpose of which was to remove all the little colonial anachronisms from Australian law and ensure that it would not be necessary for teh British parliament to legislate for Australia (even on request) again.
Finally, there was and in some cases still is one other form of colonial legal connection between Australia, New Zealand, Canada etc and Britain, which has to do with law courts. For many years, the highest court of appeals in dominion legal cases was a British court (the Privy Council of the House of Lords, in London). Appeals to this court were abolished from Canada in 1949, from South Africa in 1950, from Australia for federal cases in 1975 and state cases in 1986. In New Zealand, however, the right still exists, as it does from a number of other former British colonies too small to have a permanent supreme court of their own.
Update: Ken Parish points out that the Privy council is not technically part of the House of Lords. (This is in fact implied in the name - the Privy Council was originally a private council of advisors to the king, separate from parliament). The court I was talking about is the "Judicial Committee of the Privy council", which is a different court to the "Judicial Committee of the House of Lords", which is the highest court of appeal for British domestic matters. The two courts have mostly the same members, however.