Wool has been washable for centuries and if the right quality of wool is used in a bedding product then there is no requirement for it to be treated with chemicals during processing in order for it to be washable.  Obviously, as a natural product, care does need to be taken to ensure longevity. In the UK, manufacturers and producers of wool bedding have to follow strict guidelines regarding labelling to show, not just the individual components used in a product but also how to care for that product.

Good quality wool can be washed by hand or on the gentle wool care cycle of a washing machine

Wool filled bedding can be kept fresh by hanging up to air, whenever you change your bed linen. If a more thorough clean is required, then wool bedding containing good quality wool can be gently washed by hand and then spun dried or washed on the wool care cycle of a washing machine, using wool care soaps or wool-friendly detergents before hanging up to dry. Wool should not be tumble dried unless it has been chemically treated.

Washing machine - Wool Care