Northern scandinavian, here.
This depends on where specifically. Near the ocean it’s not that cold at the moment (well, depends what you’re used to, I guess. I get around fine with t-shirt and a light jacket. Temp varies between 5C and 15C nowadays.

But in general, I suggest you bring a sweater , some woollen socks, and a jacket that can handle rain/slush/snow. Staying dry means staying warm. Dressing with layers is better than one thick coat.

Snow will set in soon, though, so if you’re going to be working outside I recommend some gloves. Preferably two pairs, so one can dry off while you’re wearing the other pair.

May I ask where specifically you’re going? Coast vs inland makes A LOT of difference. Humidity can make 5C near the coast feel colder than -20C inland. Plus wind isn’t as much of a factor far away feom the ocean.

Layers are important for your clothing. You don’t want to get too hot if you’re working outside. Breaking a sweat will make you freeze, so you need to be able to adjust throughout the day.

Northern Canadian here. Your worst enemy in the cold is wetness. As others have said, layers are key. Silk and wool are top of the list, but synthetics are okay, too. Silk and wool are expensive, synthetics are cheaper. Do NOT wear cotton. Cotton gets wet and stays wet. It truly sucks in cold weather.

Sweating makes you wet. You have to match your layering to your activity. If you are going to be active, don’t overdress. You should feel chilly when you first start your activity. A common trick is to layer up, then take off your parka to do physical activity, then put it back on when you are done with the activity. Some jackets have pit zips that you can open to shed excess heat. If you are going to sweat, plan it so that you end up indoors somewhere you can dry out. Don’t sweat and then plan to stand around or sleep outside.

If you are going to be mostly standing around, you need big, bad-ass Baffin-style boots, which are heavy. If you’ll be moving around, you can use insulated hiking boots and wool socks. Bring extra underwear and socks because they get wet.

Mitts and a touque are mandatory. Bring two sets because they get wet. Gloves are much less warm than mittens. You can layer that, too. A very thin synthetic glove inside of a mitten works when you need to take off your mitts to work on stuff. It is also worthwhile to get a thin, synthetic balaclava to help prevent wind burn and frost bite. Fingers, toes, and cheeks are the most susceptible to frost bite.

Grow out your beard if you are a dude.

In terms of less intuitive tips, as someone else said, if you start getting cold, expelling urine and faeces really does help. Also, stay hydrated. You get cold when you get dehydrated. You may not even feel thirsty, but cold air is dry air and you will get dehydrated quicker than you think in the cold. Especially if you are shoveling snow.

Shoveling snow sucks, so people tend to rush. The key is to go slow, especially if you are older. You will build up heat rapidly if you are shoveling. Avoid sweating too much, unless you have somewhere warm to dry off. Even if you aren’t shoveling, manhandling a snowblower will make you sweat heavily, too.

Layers - get a good base layer, merino wool is the gold standard for these. You also want to get a neck gaiter, plus some merino wool socks, and some kind of cap. These will keep you warm and dry throughout most cold weather. Add more layers on top when outside, and remember to remove layers proactively if you feel yourself starting to sweat - it’s never a good idea to be wet in cold weather. The base layer will make physical exertion a pretty hot affair quite fast, so you’d be surprised how fast the sweat starts pouring.

money

Vitamin D supplements. You’re not gonna get much sunlight, and you need vitamin D not to get depressed.

The locals are used to seasonal depression. Foreigners tend to have a hard time with it.

If you can, ask someone already there. Depending on the exact type of work, they may have very specific recommendations, and they’ll have a better idea of what the climate is like than we will.

In general, you’ll want to be able to layer clothing. Start with the base layer, whose job is to be wicking away sweat.

Mid layers are for insulation, and it would be prudent to get one light sweater and one heavier sweater, maybe a third or whatever. The idea being you can increase your mid layers to stay warm but not too warm as necessary. Same for pants. Around here, I usually go for a tight base layer, a loser waffle-weave longjohns and shell blouses into boots.

Keep in mind, that layers being worn to the outside should be looser so as to not compress layers being worn closer to the body.

Patagonia makes some good, hard wearing stuff as a general brand to check out, but there far from the only one.

Hats and gloves are important, too and for gloves I’d consider getting mittens at least as one option and maybe lighter fingerless gloves to wear inside. (Or lighter gloves. Especially if it’s possible you’ll need manual dexterity)

Also, bring a book or something to read, and plenty of snacks that don’t necessarily freeze. Hydration is also important and illumination.

Try to stay away from caffeine, among other things it also constricts the blood vessels limiting circulation in your extremities.

I don’t live so far north, but good, light, waterproof boots. Of course, everything else people recommended already - good gloves, socks, jacket, and layers. But for me personally, boots would be the most important.

You going sometime soon? Around December the temps tend to plunge and Jan-Feb can be really cold, somewhere around -40C, potentially with some wind and blowing snow making it feel even colder. Highly recommend good outside set, with winter boots, gloves and the rest of clothing meant for that sort of weather, especially if you expect you would need to be outside for a while. I like the overall type pants and winter rubber boots big enough to fit wool socks in myself. :P Might be difficult to buy good quality stuff if you live somewhere much warmer. Probably worth asking whoever is hosting you for some help there. They might even have extra sets available already. Houses up there are typically very warm, so nothing special needed for inside use. Otherwise maybe a camera that is good in low light if you like to take pictures. Should get some good northern lights and stuff if you’re up there a whole month.

spikes, wool, layered clothing, reflective vest or other reflective gear…

Layers are the way to go.

1. Long Underwear, a long sleeve breathable undershirt, good WOOL socks
2. A sweater or other medium-thickness top garment.
3. Waterproof snow pants and coat.

Good boots, preferably waterproof if you’ll be in the snow a fair amount. Definitely good so you have better grip on snowy/icy ground. Thin “liner” gloves and thick waterproof gloves to go over those. A warm hat that covers your ears (I really appreciated my trapper hat when I went to Tromsø).

Avoid cotton. Cotton is horrible at insulating if it gets wet, as opposed to wool which can keep you warm while fully drenched.

For the outer layer, do not go with synthetic stuff, despite what people will tell you. An oversized Icelandic wool sweater or similar will catch snow and wind much more effectively (and dry in no time when indoors.

Gin and tonic

There’s a lot of good advice here already, especially that wool is the gold standard - nothing synthetic cuts it. I want to add that the absolute key is about layering, and not over-stuffing.

What keeps you warm is primarily the air trapped between your layers, which means that three thin layers can be a lot better than one thick layer. This also means that you will be freezing if your layers are too tight. If you have two thin layers, and put on a sweater, and that sweater feels tight, that likely means you’re pushing out the air trapped in your inner layers, and they won’t be as effective. The same applies when putting on a jacket.

So: You want a thin base layer (think light, thin wool shirt + long johns), then an optional medium layer or two (slightly thicker wool shirt, I have some in the range of 200 grams), and finally a thicker sweater for when you’re not moving. These should increase in size so that they can fit the thinner layers underneath, and you want your jacket big enough to fit all the underlying layers.

Finally: When you’re moving around, you will get stupidly warm and sweaty unless you take off clothes. It’s better to take off some stuff and be a bit cold for the first 10 minutes of moving than to get sweaty and be cold for the rest of the day. If (when) you do get cold, running in a circle for 10 min will fix it (run at a calm, steady pace, if you’re really cold it might take longer to get warm than you think, but you will get warm if you move).

In short: Being in a cold climate is just as much about how you use your equipment, and how you activate yourself to stay warm, as it is about what equipment you have.

Your wife and young son, plus a sturdy axe.