Natural mattresses apart from being better for the environment are selected because "foam is regarded as the enemy." Foam is petroleum based, has a chemical smell when new. It raises questions about the chemicals used in fireproofing the foam.
Foam can be replaced by coir and lambswool or coir or lambswool and pocket springs - that's straightforward.
You have a dilemma when choosing the cover:-
A 100% natural fabric like cotton will be breathable but will NOT be waterproof, so clearly not Ideal for a baby mattress.
There is a three layer fabric called Outlux that is 92% cotton with a thin layer of PU in the middle to make it waterproof but it is NOT very breathable because the only air that gets through is from the thin layer of cotton above the PU. However you might accept this to get as near to a natural a fabric as possible.
Probably the best option is to protect the natural base with a separate waterproof cover that has highly breathable EasyChange toppers that zip on to it. Easy to zip off the soiled topper and zip on the clean one. These covers will be waterproof and fully breathable but NOT 100% natural.
NB: The sheet that your baby sleeps on can of course be 100% cotton.