Specifically, what powers does the British Monarch have in these nations? Does the Queen have veto power over potential canadian laws? Can Australians be knighted?
In Canada, the governor general is the Queen’s personal representative. A bill becomes law only when he proclaims it. Theoretically, he could refuse on instructions from the Queen. IMHO, this would be the end of the monarchy in Canada. Being a member of the Commonwealth confers no such power on the queen, however. Recognizing the queen as sovereign is what gives the GG that power. I believe that India is a member of the Commonwealth, but is a republic.
The British monarch has no powers in Canada.
The Queen of Canada has those powers carried forward by the Constitution of Canada.
This is not simply a semantic difference. The British monarch takes her advice from the British Prime Minister and no-one else.
The Queen of Canada takes her advice from the Prime Minister of Canada and no one else.
So are: Cyprus, Cameroon, Namibia, Samoa, Fiji, Rwanda, Zambia, Maldives, Singapore, Botswana, Nauru, Bangladesh, Mozambique, Seychelles, Dominica, Kiribati, Vanuatu, Ghana, Guyana, Kenya, Malawi, Malta, Mauritius, Nigeria, Pakistan, Sierra Leone, South Africa, Sri Lanka, Uganda and Trinidad and Tobago
So are: Cyprus, Cameroon, Namibia, Samoa, Fiji, Rwanda, Zambia, Maldives, Singapore, Botswana, Nauru, Bangladesh, Mozambique, Seychelles, Dominica, Kiribati, Vanuatu, Ghana, Guyana, Kenya, Malawi, Malta, Mauritius, Nigeria, Pakistan, Sierra Leone, South Africa, Sri Lanka, Uganda and Trinidad and Tobago
Right. Being a member of the Commonwealth is sort of like having the Queen as your big sister. There is somewhat of a social expectation that you will look up to her as a role model and whatnot, but if you choose to disregard or “disobey” her, the most she can do is stop inviting you to family reunions.
What would happen if the GG refused to follow the instructions of the Queen? Treason?
No. Basically it would never happen. My take on it is that the principle of a sovereign responsible government would prevail, a principle that says that the Queen of Canada can constitutionally act for Canada only on the advice of her Canadian ministers.
So although there is some very remote theoretical possibility that a GG might decline to give Royal Assent to a bill for a number of very unlikely reasons, the possibility of the Queen overruling the GG is a non-starter. Any such decision made by the Queen alone in defiance of the GG would likely be held by the courts to be constitutionally invalid and would spell the end of the monarchy in Canada. More realistically, the Queen would never do this.
The Queen’s Privy Council in the UK remain the highest level of legal appeal (for now at least) in many Caribbean countries.
Which has the interesting side effect that the UK courts are still hearing death penalties appeals even though the UK abolished the death penalty 60+ years ago.
What if that advice is conflicting? UK PM: “Brenda, you should wear more hats; makes you taller.” C PM: “Mum, leave the hats behind; they make you look old.”
Since they are separate countries, the advice won’t conflict. What she’s advised to say and do as Queen of Canada is not London’s business, and vice versa.
Then Her Majesty will be guided by circumstance. In this case, the course of action is obvious. Her reply to David Cameron should be: “Stick it in a sow’s orifice, you gormless ponce, and don’t call me Brenda.” Her reply to Justin Trudeau: “I am old, sweetie, and more’s the pity or we’d have some rumpy-pumpy back at the old pile.”
Then in the UK, she’d wear the advised number of hats, and when in Canada, she’d go hatless.
As she spends most of her time in the UK, it would impact her little 
In addition to the Royal Assent and conferment of honours Governors-General also exercise the perogative of mercy (issuing pardons) but only on the advice of the Prime Minister. It’s pretty rare in New Zealand, (I can’t think of any since Arthur Allan Thomas).
There are also the reserve powersto act without Ministerial advice in exceptional circumstances, but no NZ Governor-General has ever used them.
Why? Given that the governor-general is the queen’s representative, why can’t she just step in and do things herself? I can’t imagine that would go against any protocols. If I recall correctly, there was a time when the queen was in Canada, and gave royal assent to a number of bills in person, instead of having the governor-general do it.
(That said, if the governor-general refused royal assent, I would wager that what would happen is that he or she would just be replaced with someone who would grant it.)
The only constitutional power the Queen of Australian has is to appoint the Governor General on the advise of the Prime Minister.
Australian can still be knighted, but under the British Honours system and on recommendation by the British government.
The Australian Honours system was commenced in 1975 and did allow for the creation of Knights and Dames until 1986. There was a legacy provision for Australian State governments to recommend imperial honours until 1989.
In March 2014 PM Abbott reintroduced knighthoods to the Australian Honours and in Nov 2015PM Turnbull removed them again.
Royal Assent is one of the powers that either the GG or the Queen can exercise. In Canada, the Queen almost never does – the major exception was the signing of the Canada Act of 1982 patriating the Constitution, in which the Queen’s visit and signing the Royal Assent to the Canada Act was merely a symbolic and historic moment.
Conflicts between the Queen and the GG cannot occur, in practice, for the reasons I already gave. The term “responsible government” isn’t just a catchphrase but a specific principle in governance:
Responsible government refers to a government that is responsible to the people. In Canada responsible government is more commonly described as an executive or Cabinet that is dependent on the support of an elected assembly, rather than on the monarch. It originated in Canada in the 1830s and became an important part of Confederation. It’s the method by which Canada achieved independence without revolution.
Responsible Government | The Canadian Encyclopedia.
I’d love to take you on that wager! 
The Prime Minister can advise the Queen to overrule or dismiss the Governor General; under normal circumstance Her Majesty would obliged to accept such advice.
Some of you seem to think that the Queen is like the Queen of Hearts in Alice in Wonderland - “Orf with their heads…”
The truth is that she and her family are this country’e greatest diplomats. For whatever reason, they command respect all over the world, even from diehard republicans. Maybe because they are a symbol of stability in an ever changing world; that they are determinedly apolitical - I don’t know. But whatever it is, we can be 100% confident that they will not abuse the privilege.
Even when they are virtually at war, Commonwealth countries like India and Pakistan can sit at the same table with the Queen at the head.
It’s helpful to think of the monarch and the various governors general as sort of national notaries. (I don’t think I came up with this myself. I think I read it in a previous thread.) Royal assent simply means something like, “the government has followed all the proper procedures and filled out the proper forms for this act to now be considered official.”
What would happen if the Queen or her representative were to refuse to do their job of certifying an act that had properly been passed by the relevant Parliament? The same thing as happens when anyone else refuses to do their job: they’d be replaced, their position would be abolished, or if those were politically infeasable, people would simply find a way to work around it.