Linen vs. cotton was addressed previously here, which I think answers part of your question. (Although I would correct myself - printed cottons were edging out linen by the 1780s, not in the early 19th century.)

Wool is a little more difficult, because it doesn't have the same kind of simple "first wool, than cotton" narrative as linen. Linen for outerwear had a fairly narrow range of options for quality, but wool could go from coarse and fit only for paupers to being woven in a damasked or brocaded pattern, blended with other fibers like silk, or spun into extremely fine threads. As printed cotton became cheaper, yes, it eventually took over one market, but wool continued to be necessary for coats and cloaks, flannel underclothing, mourning dress, and was still used for nicer day dress. Wool and silk continued to be relatively common fibers despite being more expensive than cotton until the development of regenerated and synthetic fibers like rayon, dacron, polyester, etc. in the mid-20th century. These could be manufactured and thus sold very cheaply, and mimicked the properties of wool and silk well enough to satisfy consumers looking for the lower price point.