Stone, Mortar, and Limeplaster
They live at the foot of a mountain... a small river runs near the town
As other questions have already pointed out, local stone is much easier to gather and stack, than wood is to chop down and carve, and this river will give the the best kind of stone for stacking. Anywhere you have a river that runs through a mountain, you get plentiful naturally occurring, ready to use stone. The faster flowing waters of mountain rivers wash away smaller sediments and leave behind riverbeds full of smooth stones that are just the right size to be used without needing any shaping at all. Just go out to the river with a wagon, load up, and come home with plenty of building materials.
While some slower moving parts of the river may produce silt and clay, these will likely be the exception more than the rule due to the proximity of the mountain.
In the Medieval period, the technology to make mortar was very well known. There were 2 general types: Ash and Limestone. Ash mortar was cheaper because all you needed for the cement was wood ash. Since every house was heated by and meal cooked by wood burning, ash was a very plentiful resource that was recycled into many things including mortar. Mixed into a mud it will cure to form a low grade concreate that will not dissolve in water like mud would, but is more prone to cracking from normal ware and tear than limestone mortar.
The higher quality mortar is made by heating limestone until it is red-hot and then dosing it with water which creates a chemical reaction forming quicklime. When mixed with mud, this creates a high grade mortar that can easily last for hundreds if not thousands of years. Limestone will be very common in a place like this; so, if you factor in that magic is used for heat instead of burning wood, then this may easily be the most used kind of mortar since you wont need to burn a lot of wood to get it up to heat, and wood ash will be less common with fewer reasons to burn wood in bulk.
Wood will only be used for widely used for roofing and doors, and maybe for windows. Another thing of note is that a lot of medieval architecture did not look like it looks today. Home owners would often coat the outside of both stone and wooden buildings with a lime plaster made of the same material they used for mortar, to give thier walls a smooth white finish. So most of your middle-to-upper class homes should be distinguished from your lower class stone and mortar homes by having a stucco looking façade instead of bare stonework.