The glossopharyngeal and vagus nerves are usually tested (and described) together because isolated lesions of one but not the other are essentially unknown. Obvious features would include an absent gag reflex and a uvula deviated away from the lesion (these were demonstrated in Question 21.2 from the first paper of 2014). In the awake patient, a hoarse voice may indicate laryngeal paralysis. In the intubated patient, the cough reflex is a way of testing the vagus nerve (it forms the sensory component). In the CICM fellowship exam, the candidate will be one among about twenty, and everybody will want to give the patient a thorough Yankeur-suckering; in view of this the examiners seem to frequently tell the candidates not to perform this test.

Lesions of the glossopharyngeal and vagus nerves

Obvious features

  • Absent gag reflex
  • Deviated uvula (away from the lesion)
  • Laryngeal paralysis (unilateral or bilateral)

Subtle features

  • Difficulty swallowing
  • Impairment of taste over the posterior one-third of the tongue and palate
  • Loss of sensation over the posterior one-third of the tongue, palate, and pharynx
  • Dysfunction of the parotid gland

and, via the vagus...

  • Decreased oesophageal motility
  • Dysregulation of heart rate

Causes of a unilateral CN IX and X lesion

  • Tumour (cerebellopontine angle, acoustic neuroma)
  • Trouble in the jugular foramen (eg. base of skull fracture)
  • Medullary infarct
  • Carotid artery aneurysm
  • Recurrent laryngeal nerve paralysis due to a neoplastic mass or surgery along its course

Causes of a bilateral CN IX and X lesion

  • Pseudobulbar palsy
  • Medullary infarct
  • Bihemispheric infarct (supranuclear)

Bulbar palsy

Question 27 from the first paper of 2019 asked for the lesions of bulbar palsy, and their coresponding cranial nerves. There was nowehere else to put it, and so it remains here, even though the glossopharyngeal and vagus nerves are not the only ones involved.

Basically, bulbar palsy involves nerves IX, X, XI and XII, which makes little sense because this palsy is bulbar, and therefore involves all things of the bulb, which is an archaic term for the medulla oblongata and occasionally for the brainstem as a whole. Why we persist with this rather than calling it medullary palsy is not entirely clear.  In extension of this, it is unclear as to why we still use the archaic term "palsy", as it appears to take its origin from the Latin paralysis, mangled by centuries of illiterate Anglo-Normans.  

Anyway.  According to LITFL and the CICM examiners,  the clinical features of bulbar palsy are pure lower motor nerve signs of the medullary cranial nerves. Kühnlein et al (2008)  is probaby the best description of these features writing about ALS patients:

Absent gag reflex: due to bilateral lesions of CN IX and X

Soft palate paralysis which results in "im­precise consonants, hypernasality, and a decreased range of pitch and loudness" (because the weakness of the soft palate and pharyngeal muscules causes insufficient nasopharyngeal closure and reduced oral airflow). The resulting speech is breathy, quiet and the patient is generally unable to generate long phrases because they need to pause to take another breath. This is due to bilateral lesions of CN IX and X; a unilateral lesion of the same mechanism would cause a deviation of the uvula.

Paralysis of the laryngeal muscles: this causes a "soft, weak, low-pitched and
mono­tonous voice" 
which is the result of CN X paralysis. In bulbar palsy the volcal cords are hypoadducted, which promotes aspiration.

Tongue wasting and fasciculations due to CN XII paralysis; which also contributes to the inarticulate speech.

Dysphagia occurs because the process of swallwoing normally requires all the bulbar nerves to work in concert, and their weakness therefore results in an incoordinated swallow. Realistically, this is due to the dysfunction of nerves IX, X and XII, but also V and VII.

Drooling, because there is a loss of tone and strength in the muscles that control lip closure (CN VII). At the same time, due to decreased laryngeal sensitivity and motor function, there is progressive pooling of food and saliva in the vallecula and piriform recesses.

References

The LITFL summary of cranial nerve lesions is without peer in terms of useful information density.

Walker, H. Kenneth, W. Dallas Hall, and J. Willis Hurst. "Clinical methods." 3rd edition.(1990). Chapter 63. Cranial Nerves IX and X: The Glossopharyngeal and Vagus Nerves- by Kenneth Walker

Rea, Paul. "Chapter 5: Hindbrain (Rhombencephalon)" Essential clinical anatomy of the nervous system. Academic Press, 2015.

Kühnlein, Peter, et al. "Diagnosis and treatment of bulbar symptoms in amyotrophic lateral sclerosis." Nature Reviews Neurology 4.7 (2008): 366.

Graham, P. J. "Congenital flaccid bulbar palsy." British medical journal 2.5400 (1964): 26.