Praise the sun.

Just like before compasses existed, the sun is your guide. It rises in the east and sets in the west. At midday or night, you will have to remember which direction is which or wait until you can see the sun's direction. Additionally, you can use charcoal to update your map with stuff that you can see.

Otherwise, you'll just have to study the map before you set out and hope that you can find landmarks along the way.

... and drop a stick.

On Reddit 3 days ago TheLedVette said

Drop a stick. I know it seems so simple but when you drop something it will always fall the same direction.

Seems to work! A dropped stick points consistently north in ML at least.

For now at least, maybe this will change in the next release.

There are several ways to navigate/orient without a compass. As mentioned above, the Sun rises in the east and sets in the west, but you don't always have a clear sky and obviously this is of limited utility at night.

Another method, which is important, is landmark spotting and using waypoints. There's no automatic waypoints in TLD, as you note it runs counter to the game's theme, but that doesn't mean you can't use your own. Learn the landmarks around your home: The railroad is north of the main lodge on Mystery Lake. The grain silo is East-by-Northeast of the farm house in Pleasant Valley. Look at the map and figure out what landmarks you expect to see as you move along and if you don't come upon those landmarks, it's time to re-orient using the map.

Bad weather will absolutely disorient you - it will also kill you. Two very good reasons to hunker down when the blizzard comes in. Even heavy fog is bad news very quickly - even if you don't get lost you often won't see the bear or moose until they've seen you... and then it's too late.

Spray. Paint. You can spray arrows for trailmarkers, you can tag things in ways that show up on your map as long as you tag them in relatively clear conditions, etc. This is super important in caves where there is no map.

Finally, charcoal is your friend. First off, it's how you even have a map. Secondly, even if you've fully mapped the region, you can still use the charcoal to locate yourself on the map. You may consider this cheating, so make that choice first, but when you use charcoal, a circle (lighter in color than the rest of the map) will radiate outward from your actual position on the map. Bam. You know where you are in that moment.

If you absolutely must be out in bad weather, or are planning long-range treks. Make sure you've got at least 5 cloth on you. That + sticks = blizzard shelter, enough to warm you up even in the worst storm and get you through to clearer weather or give you time to consult the map and figure out a plan.