If you want no wool or synthetic, your choices for clothing are quite constrained. In materials you are left with plant fibers; cotton, linen, hemp, or you can go semi-synthetic with rayon, bamboo, tencel, lyocel, viscose, etc (broadly, all are variations of viscose)

Excluding wool/synthetics is not necessarily better for the environment. Some examples, socks usually contain some nylon or similar to help with durability. Arguably a bit of nylon in your socks is better than wearing through more socks more quickly. Viscose, rayon bamboo, etc, are often greenwashed and marketed as green alternative fibers, in reality making these fibers is a toxic chemical process executed in third world countries where the environment gets destroyed as a result. Cotton requires much water and pesticides.

Dressing for the cold or wet will be hard without wool or synthetic

Ideally I recommend you buy only what you need, and try to find european/american production, or shop from reputable brands, also secondhand shopping and repair what you can.

Conscientious shopping is a great step in the right direction