PUL is appropriate to use for the waterproof bottom. It can be machine washed and dried - in fact, machine drying at least occasionally is advised for PUL in order to seal any tiny holes that may have appeared in the lamination due to use or accident such as diaper pins or what have you.
Another option would be water resistant regular or ripstop nylon. These can also be machine washed and dried.
For the top layer you can use bamboo or cotton fleece, sherpa, hemp fleece, terrycloth (2-3 layers at least), or even an old towel or two cut to fit your needs. Wool felt would work well also. Another option would be flannel, though it would also take more than one layer; at least 2-3 layers would be more effective for your purpose.
You are correct - the synthetic fabrics I know of are not absorbent. In fact, polyester fabric is non-absorbent and you can read that anywhere on the web. There is one major exception - microfiber. You can find it in the auto section of big box stores or in automotive stores. They absorb well and are an inexpensive alternative to natural fabrics. The problem with microfiber is you need to remember to keep it from touching delicate skin since it draws moisture so well.