Car wobbles at low speeds

When my son bought this car the previous owner had two snow tires on front and two regular tires on the back. He gave us the other two regular and two snow tires. Everything felt fine until my son switched the front tires from the snow to the regular tires. Now at very low speeds the car wobbles. It feels like the wobbling is more in the back than the front.

I brought it to a mechanic, for a totally different reason, and asked him to balance the back tires (I was mistaken. I thought it was the back tires that my son changed over). The mechanic said the tires are bad (metal is sticking out) and that's why I'm getting the wobbling. It all made sense until my son pointed out that, like I said, it wasn't the back tires that got changed, it was the front.

So now nothing makes sense to me. If he just changed the front why does the wobbling feel like it's coming from the back? Could it just seem that way but it's really coming from the front? Also, at first I only felt the shaking in the car itself, not in the steering wheel, when going about 5-10 mph. Now at about 40 mph I'm feeling it more in the steering wheel. Also, I don't know if it's my imagination but the car feels a bit bouncier now.