This excellent article from New York Magazine focuses primarily on the breathing aspects of the vagus nerve and the mechanism behind the calming affects of slow breathing.
“Stimulating the #vagusnerve to the heart has a really powerful effect on slowing the heart rate,†said Lucy Norcliffe-Kaufmann, associate professor of neurology at NYU-Langone. And this, specifically, is what relaxes us. The vagus nerve is basically listening to the way we breathe, sending the brain and the heart whatever message our breath indicates. Breathing slowly, for instance, reduces the oxygen demands of the heart muscle (the myocardium), causing our heart rate to drop.
When in a stressful situation and anxious about how to respond, consciously slowing your breathing for one minute, or even a few seconds, can put yourself in a calmer state, enabling you to process and communicate better.
Source: https://www.thecut.com/2019/05/i-now-suspect-the-vagus-nerve-is-the-key-to-well-being.html