"Will meat and dairy be outlawed in the future? What will live in the rural arctic be like when veganism is the law of all the lands?"
No meat and dairy will not be outlawed, at least anywhere in the near upcoming future. Also in places such as the rural Arctic a strict vegetarian diet would be a very expensive lifestyle. No one can prognosticate what the law of the land will be in the future, although the potential possibilities can be considered. Also there are two things that simply can not be regulated, or at least for any length of time. The first is freedom of choice, that while it has been regulated or more accurately attempted to be regulated, which historically speaking, has met with dismal failure. The other is the freedom of thought, which while it could be attempted would not succeed, unless one belongs to the tin foil hat group, that believes that thought is regulated and controlled, and are also often believers in a multitude of conspiracies.
Also there are many but not all vegans, who fail to realize that without some hunting, the strict vegetarian diet will not be sustainable. That I take from something I heard many years ago, from a biologist that was said by a vegan. He stated to a young vegan, that in order for the strict vegetarian diet to be a sustainable diet, that a percentage of the human population would need to continue to hunt and eat meat. Now his I.Q. was much higher than that of the then young vegan. Then there is the fact that there isn't a singular one size fits all diet. That fundamentally I learned from a vegan who was both a registered nurse, and was also a licensed dietitian. Now of course there are vegans, and even some vegetarians who without the degrees held by either of those two, would argue and very strenuously that they're wrong.
Now as correctly conjectured, in another answer given, it won't happen for a few hundred years, if it happens. Also it's been proven that the outlawing of things has many times met with abysmal failure. But yet there are always some, try and trivialize those failures, by saying it would have worked if more time had been allowed to pass. Now as for the two sides, one saying it will happen, and the other who say it won't happen, their crystal ball works no better than what mine does. So it all comes down to nothing more than speculation, as to what the future holds. While we can look to the past for guidance in some things, then conversely in other things we can't, as there's not the precedence there to use, or even review.