If your neck breaks and you lose control of everything below one of your cervical vertebrae, how do all your internal organs still work? Your heart, spleen, digestive system, etc. all still work because if they didn’t, then a cervical break would be death for anyone who has had one.
If you suffer from psychological panic, does your heart race still increase?
Does stress still cause your digestion to become dysfunctional?
Do you get an erection if you see something stimulating?
Well, there are two separate communications systems in your body, your nervous system and your endocrine system. Communication that doesn’t need to happen very quickly or require much detail tends to be handled by your endocrine system. So, glands someplace secrete hormones into the bloodstream, and organs elsewhere adjust their operation accordingly. That’s what hormones DO. Some of what you are asking about is endocrine.
If the cervical injury happens high enough up in the neck (say C1 - C4), you may not be able to retain life absent a mechanical breathing apparatus.
The vagus nerve has a regulatory role in several systems… It is the tenth (X) cranial nerve, and it arises from the medulla oblongata and not the spinal cord. It runs from the head all the way down to the abdomen, with its branches innervating multiple tissues along the way. The vagus provides some regulatory function to the heart, stomach, and intestines.
An injury to the spinal cord will not necessarily sever the vagus.
Some things, like erections, have a reflex component. Men with breaks in their cervical spine usually have that erection reflex system intact, as it’s much lower down in the spine. Men with breaks in the spinal cord lower down are more likely to have that portion of nerves messed up, in which case they might be permanently unable to achieve an erection - but even so, that varies from one man to another.
Some of the rest of what you talk about - reactions to stress, for example, and digestion - likewise either involve nerve systems that would not be affected by a cervical break, or are run by hormones which, likewise, would not be affected by a damaged spinal cord, or a combination of both.
And, as noted, the vagus nerve does not run through the spinal column and thus its functions will not be impaired by a cervical break.
On the flip side - it is possible to damage the nerves controlling, say, your intestines without impacting those in your spine. This can lead to things like megacolon and Hirschsprung’s disease. So your limbs work just fine but you can’t, as the polite put it, “void waste”. The condition can be fatal and the Mutter Museum as an exhibit of one such organ on display.
Basically, the system that runs your innards and the system that allows voluntary movement are two different systems. Damage to one in unlikely to affect the other significantly.
Also - the human heart is quite capable of beating independently, even when outside of a human body entirely (though obviously it won’t do so for very long without a steady blood supply). Transplanted hearts also work fine, despite all nerves being severed. The pacing for the heart is set within the heart itself (except when it goes awry and an artificial pacemaker is installed). Severing the nerves between the heart and the rest of the body does have some effect - if I recall, transplanted hearts tend to beat faster than non-transplanted ones - but they are not incompatible with life or health.
I know from a relative injured in a motorcycle crash that someone can b confined to a wheelchair, but he is able to talk, work, etc. His kidneys still worked and produced urine from what he drank, but he didn’t have voluntary control of his urination – he had to wear a catheter & urine collection bag.