I am not familiar with cholinergic neuropathy, but, mestinon is supposed to increase the effect of the acetylcholine (ACh) that is released by the neurons.
ACh is used in the autonomic system, and in the innervations of the muscles.

The neurons release ACh in their synapses with the muscles fibers, and when enough ACh reach the receptors of the muscles, the muscle constricts. ACh esterase clears this ACh from the synapse. Mestinon inhibit the activity of the ACh esterase, so it increases the effect of the ACh, because it stays for a longer time in the synapse.
So at the bottom line mestinon increase the muscles constrictions. (So it seems weird for me that it was given for a spasm, but I don’t know…)

Mestinon is usually used in myasthenia gravis, in which there are antibodies against the ACh receptors in the muscles fibers. So helping the ACh stay for a longer time in the synapse supposed to compensate for the damaged receptors.

Regarding the autonomic problems, ACh is involved in the sympathetic (Fight or flight response – faster heart rate, higher BP, increased sweating, etc) and the parasympathetic (rest and digest activity, such as tears, salivary, urination, lower HR and BP, etc.) branches. From the little I read cholinergic neuropathy affects mostly the postganglionic neurons in these pathways so it affects all the parasympathetic innervations, and the sweating which is sympathetic. So mestinon is supposed to increase tearing, salivation, sweating etc.