Imaging. Modalities such as CT would provide good 3D information in a rapid manner. An MRI would also be an excellent choice for the superior soft tissue definition, but is more time consuming and costly.

A plain film x-ray is possible, but probably will not show enough detail compared to a CT/MRI. Ultrasound is also possible, but again the CT/MRI will provide superior detail.

Endoscopy - An upper endoscopy provides an excellent view of potential strictures. Alternatively, imaging with an barium swallow study (x-ray taken after drinking a barium contrast solution) would also show potential narrowing. Barium swallow study is particularly useful in patients who many not be able to tolerate endoscopy well.

Pharynx deviation while swallowing - The action of the vagus nerve on pharyngeal muscles elevates the pharynx. Since the muscles are elevating, the pharynx will deviate to the intact side.

Tongue deviation on protrusion - The action of the hypoglossal nerve is on tongue musculature. When protruding the tongue, the intact half of the tongue will deviate the tongue towards the weak side. 

Gag reflex - The motor function of the glossopharyngeal nerve is minor and often cannot be readily visualized by pharynx deviation. So the most reliable test of function is the prominent pharyngeal sensory role, where touching a q-tip to each side of the pharynx to elicit a gag reflex tests sensation. The impaired side will have diminished or absent gag reflex.