Thanks to Lhipenwhite from Absolute Write for the excellent question.
I went to the barn and did a little bit of experimentation. Generally, you do use both hands to mount. However, with practice it would not be that hard to mount with only one hand - providing the hand you had is the one closest to the horse's neck (otherwise you couldn't hold the reins).
So a rider who had lost his right hand would need some practice learning how to mount again, but could manage (except possibly with a giant horse). A rider who had lost his left hand would probably need to mount from the right, normally incorrect, side of the horse. Definitely a detail worth remembering.
Additionally, it is actually harder to learn to dismount without the use of one hand, again, especially on a large horse. Dismounting would be a struggle...but with practice, again, the rider could adapt.
The horse would also need to be trained to neck rein so it could be easily controlled with one hand.