So when it comes to cleaning the mitt and towel, it will obviously need to be washed after each use. I'm guessing just a 30 degree cool wash is best?
No, not if you want your mitt to last as long as possible. That wash will be fine for the towel or micro fibre cloths but definately wash them separtely as they do run.
To prolong the life of your lambswool mitt you need to hand wash and rinse it, then gently squeeze out excess water, reshape it and hang it up in the mesh drying bag, somewhere warm-ish (don't put it on a radiator) and let it air dry naturally. Give it a brush out before next use. I've previously washed lambswool mitts in the machine on 30°c cycle or a hand wash cycle and it has considerably shortened the lifespan of the mitt.
It's sounds like a lot of effort but really takes only 5 mins to wash/rinse out then hang it up and it'll be dry for next time