It is still cold de cold outside

I really need a new winter coat, because its still freezing cold outside, and my old winter coat’s zipper is broken, so I can not zip it up any more .So now, I have found 2 options ,both from COS.They are both really classic scandinavian and really COS, which I really like, when its comes to winter coats.The only problem is, that my mother keep saying , that they are not really worm.They are just really pretty and looks good on.But I disagree here, they are both made of wool, and wool is worm right?
wool is a worm material.I have pants made of wool , and they are really hot to wear, but good to wear when its cold outside.