Corticobulbar fibers (you can voluntarily swallow!) to nucleus ambiguus are BILATERAL (both crossed and uncrossed). Therefore, muscles supplied by the nucleus ambiguus are NOT noticeably weakened in the event of unilateral lesions of the corticobulbar system (i.e., in the motor cortex). This means that there is NO deviation of the uvula following cortical lesions. Don't confuse the results of lesions of the corticobulbar projection to nucleus ambiguus with lesions of nucleus ambiguus! Also don't confuse the bilateral corticobulbar input to nucleus ambiguus with the primarily CROSSED corticobulbar input to the HYPOGLOSSAL nucleus. What it boils down to is that BILATERAL corticobulbar input is GREAT for you as students, since you don't have to remember which way something deviates following its interruption. It is only those corticobulbar projections that are not equally bilateral (so far only that to the HYPOGLOSSAL, but more to come) that you need to worry about.
Motor fibers of C.N. XI that arise from the nucleus ambiguus join the vagus outside of the skull and innervate muscles of the larynx. These fibers comprise the CRANIAL branch of C.N. XI. REMEMBER: CRANIAL XI=AMBIGUUS. In contrast, the SPINAL portion of C.N. XI consists of motor axons whose cell bodies lie in the lateral part of the ventral horn of the first five or six cervical SPINAL CORD segments. The axons of these cells pass dorsal and laterally (that is they do not exit via the ventral root), leave the spinal cord between the dorsal and ventral roots and unite to ascend in the spinal canal to enter the skull via the foramen magnum. They then exit the skull via the jugular foramen along with cranial nerves IX and X and eventually innervate the sternocleidomastoid and the upper fibers of the trapezius. REMEMBER: CAUDAL XI=SPINAL CORD.
Lesions involving C.N. XI fibers to these two muscles result in atrophy of the muscles. Since the RIGHT sternocleidomastoid rotates the head to the LEFT (opposite), a lesion of the RIGHT C.N. XI will result in the chin being turned slightly to the RIGHT (paralyzed) side, especially when the head is flexed. The same RIGHT side lesion will result in paralysis of the RIGHT upper trapezius and slight sagging of the RIGHT shoulder.
As for cortical input to the cells of origin of the spinal part of XI, you are lucky since it is bilateral.