The area in Alaska was Tundra (No trees).I think without gas or diesel the people of Alaska and remote areas of Canada would have to go back to heating/cooking with wood. For travel they would use sleds, or kayaks. I don't agree that the knowledge has been lost. There are many who still use sleds and kayaks for recreational purposes. I would imagine that most have fireplaces or woodstoves in their homes.
The worst thing that will happen without fossil fuels is the lack of jobs and lack of food. If grocery stores cannot be resupplied, I don't think there would be enough food for the population that currently lives there. Life in the arctic would become so difficult that a huge percentage of the population would migrate south.
I didn't see the show, but if there is nothing but snow and ice, I would assume it would become uninhabitable. Good reason for those living in tundra to stock up on fuels and all other supplies if they plan to try to stay; also to have a way to GOOD if it comes to that. If there was some hunting to be done there, it would have to be hunting parties which go out from a base camp in the warmest months. The lifestyle would only be for a select few.
This is one of the rare cases where bugging out is probably the only option for many of them unless they have mastered the traditional old skills and can almost immediately adapt to that lifestyle.They've developed a lifestyle dependent upon petroleum. It's not sustainable without petroleum.
Think about how the indigenous peoples lived before the technology; small homes, igloos, etc. Their homes were not elaborate nor large, because heating them was difficult.
The carrying capacity of the region was not huge, and thus wouldn't and couldn't have supported large populations.
Take away the petroleum and people will have to move or die.
Indigenous people who traveled the tundra had evolved the ability to eat raw meat for a reason. The fuel they did use for fire was driftwood, dried plants, animal fat, and Caribou sh-t.