Canada does not have a law of royal succession. When the British modernized their law, Canada didn't follow suit because of the perennial aversion to reopening the Constitution. We simply say that whoever holds the office of British monarch is automatically, ex officio, our monarch, And what that means is that anyone who is not the British monarch doesn't have any clear legal status in Canada anymore."
So as far as I see it, if they want to live in Canada, they would have to apply to immigrate to Canada just like anyone else. But if they were will to invest very heavily in Canada, by starting a business and employing Canadians there is a possibility that they could jump the queue in immigrating to Canada. Or since they have money they could just live in Canada for 6 months and then 6 months in the UK.
We also do not recognize titles in Canada. So in the UK Harry would be Prince Harry, but in Canada he would be just Harry.
Their decision opens up a whole can of worms as to who pays for their security. But since Harry wants to live as a regular person, he would have to cover his own security. That could pose additional difficulties, since private security guards and bodyguards in Canada are generally not permitted under federal law to carry sidearms.
I would not want to be in their shoes.
Below follows a short treatise on Canadian constitutional law, particularly as to how it applies to the laws of succession for the Monarchy of Canada. If you think this will bore you to death, at least you have been warned!!
Chrisky, that is factually incorrect. Canada does have a law of succession and always has had. It was originally laid out in the British North America Act, 1867 (now the Constitution Act, 1867) and subsequently recognized in the Statute of Westminster of 1931 and the Constitution Act, 1982.
Your comments are correct only insofar as the rules have mirrored those of the UK and the other Commonwealth realms. The fact that they mirror and refer to legislation in the UK does not mean they do not exist, nor that we do not have control of and the ability to change the rules. The Supreme Court of Canada has upheld the legal right of the Parliament of Canada to change the rules without a constitutional amendment, since they already form part of our constitution.
The first time we passed a change to the rules of succession in Canada's own right, again in concert with the other Commonwealth realms, was under the authority of the 1931 Statute of Westminster. When Edward VIII abdicated in 1936, the Parliament of Canada passed the Succession to the Throne Act, 1937. It recognized the Act of Abdication by Edward VIII and proclaimed George VI as monarch.
The most recent change was passed by the Parliament of Canada under the Succession to the Throne Act, 2013. It changed the rule of primogeniture to recognize the first-born child of the monarch, without regard to sex. This was again done in concert with both the UK and the other Commonwealth realms to avoid a potential constitutional mess if we all had different rules and then ended up with different monarchs.
Also under Canadian protocol and heritage rules, HM The Queen, Prince Charles and Prince William all have personal Canadian Royal Standards and Royal Salutes, unique from their British equivalents, for when they are either physically present in Canada or performing duties as part of the Monarchy of Canada. Prince Harry has no such formal recognition, only getting a standard salute. I was going to cite the actual legislation, but Wikipedia actually does a really good job of it in more layman's terms.