The moss will have had enough time to root itself by the time the cotton has gone, but this depends on what you attach it to. I've found that wood is best, moss can attatch to this very quickly. Smooth stones or rocks will take a little longer and the moss may disattach once the cotton has gone.
I've found that black polyester thread is very good to use as it does not rot and is easily hidden by the moss as it grows.
Another tip is to tie the moss the right way down, if you look closely you will see what appears to be little brown roots on one side of the moss. Put the moss brown root side down and this will help the moss attach to the rock/wood much quicker.