My experience is the same as Jachut.
I have a wool cover that I use at night. Love it. I've had Aristocrats, and currently use a size 2 Sugarpeas side snapping cover. These are both waaaaayyy too bulky to ever fit under clothes, and even if you got clothes big enough, the bulk is too much for me. But ever since DD stopped wetting buckets at night, these have been our staple night cover. One problem I noticed is that, if I allowed her to get too wet in the SP cover (say I don't change her as soon as she wakes up, when she has her big morning pee) the urine has started to bleach out the colour a bit. I have an avocado cover but it's getting sort of yellow-green around the crotch seams. This, of course, is totally my fault for letting it get so wet, but it is another thing to consider with wool.
I now have a WIO from Luxe Baby. This is a much trimmer setup and works great under summer dresses. Because DD is a toddler and rarely sits still, we've had no compression issues. Also the cover itself has a wool doubler, and this has so far prevented any moisture from coming to the surface of the cover. I'm not sure how successful this would be with a young baby who is on his butt all day, but I'm going to find out as I'll be getting one for my new little one.
IME, with both wool and fleece, compression wicking occurs readily if the diaper can't do it's duty, and there is only so much a diaper can hold (especially if you aren't blessed with a light wetter). My DD was a fairly heavy wetter, but didn't mind a wet diaper. So it was a pain for me to rely on anything other than PUL when "out and about" since I'd have to check her all the time, and change her much more often than if I had PUL on. I have mostly ME airflows, which have never left marks on DD's legs, and never EVER leaked (not even with the messiest messes and prefolds!).
I have one fleece AIO that has never worked well as a diaper - it wicks through as soon as she wets, and it's the high quality fleece not lightweight. However, it turned out to be a fabulous summer-under-dresses diaper when she began walking, and also great for at the beach or water park (it's very pretty) so it wasn't a wasteful buy. Still, it turned me off fleece completely as a useful cover for us.
And yet, with all that said...lol...I've ordered some small wool and fleece covers for this upcoming baby. I know I'll be able to use the wool at night, and hopefully the fleece. I so wanted to give them a chance. And who knows, maybe this one won't be such a big wetter. I admit I'm envious of those for whom fleece and wool work so well. They are so soft, and nice alternatives to your basic PUL. So I'm going to give it a try again.
But you know, the PUL covers out there are so pretty. I have ordered some really beautiful combinations from Loveybums (great price!) and also Very Baby. Posie Patch snapdragons also have great prints and solids. While I know that technically, PUL is not breathable, I have never had an issue with this (mind you, all mine are side snapping, so maybe that allows more air than otherwise, who knows?).