People would scavenge for any building material they could get.
That's primarily wood to construct a shanty with. Charred beams of burned-down buildings, a broken door for a wall, worm-eaten planks of furniture as a roof. If the vegetation nearby is not completely destroyed, they would collect green wood as well.
Since big pieces of wood are hard to come by, people would build only part of their shanty above ground level and dig out the interior until an adult could stand barely upright in there.
Next are textiles. Forget fur and leather, these are too valuable to be thrown away in a still-usable condition. Cotton was historically very expensive, too. Clothes were traditionally made so they could be deconstructed and the pieces used for a new item of clothing, so the only fabrics available for the poor were threadbare rags with holes in them or gunnysack.
Due to the cold climate they would avoid any windows in a shanty. The door would be hung as thickly as possible with old bedsheets or drapes to keep the cold outside.
As roofing, people could use grass sods (as did the vikings), mud caced straw (using garbage from stables or barns) or twigs and leaves from local vegetation.
You should keep in mind that the general lack of food and hygiene and the cold winter would prevent the slums from growing to a size comparable to modern Indian slums. Most infants would die of malnutrition or infection and the old and weak would freeze to death every winter.
For the same reason, communal shelters run by charitable organizations like a church would replace those shanties. Brick buildings increase the chance of survival in winter and no honorable citizen would accept shanties constructed of garbage to spoil their city.