I am an American and am only somewhat familiar with how the Westminster system works, so I assume that there is a lot that I don't understand about how constitutional law works without a codified constitution. From my perspective, it seems to make sense for the UK, as the tradition goes back hundreds of years, but these countries are so much newer, where does their precedent come from? As I write this I'm realizing how broad a question it is, I seem to be asking both why Australia is different and how does the Westminster system work in Canada and New Zealand. Answer what ever you feel like I guess.
On another note, is there a better subreddit to ask questions on comparative constitutionalism? I often find myself thinking "I wonder why X" or "how is X supposed to work" and r/askhistorians is the first place I think to ask it, even if it really isn't a history question.
The premise of your question with respect to Canada is incorrect; Canada has had a constitution since confederation.
Originally, Canada's constitution was known as The British North America Act, 1867. Unlike the US constitution, it was a statute of a foreign parliament (in this case, the UK). In 1982, Canada's constitution was repatriated to Canada and was renamed The Constitution Act, 1867. The Canadian equivalent of the US Bill of Rights, the Charter of Rights and Freedom, was also made at this time - it is formally known as The Constitution Act, 1982, Schedule B to the Canada Act 1982 (UK).