Unilateral damage to vagus nerve
Would damage to one vagus nerve in the neck have significant impact on heart rate and gastrointestinal tract function? 


Not that much, because fibers from both sides are fused, so each nerve carries about a 50% of fibers originated from the one side and about 50% from the contro-lateral. Consider that fibers are rewired after each plexus.

References: 

 wikipedia page about vagus nerve

 wikipedia category about vagus nerve