To follow up on this question, it wasn't really anyone's job, so burials took place at random, if at all. There was no way to identify most of the dead or dying, and any markers would have been simple wooden planks.
Here's an image of Oakwood Cemetery in Richmond as it appeared in April 1865:
At the time of burial identities of almost all the dead here were known, and those involved in preparing these graves had some interest in deceased Confederates.
It would have been different behind Federal lines.
If any Confederate soldier left dead or dying along the way from Nashville to the Tennessee River received a decent burial, some very unusual and noteworthy circumstances would have to have prevailed. For example, some might have braved the winter weather and gone to the trouble of identifying a dead family member for burial. Days -- perhaps weeks -- elapsed before the search for a given soldier could begin. Difficulties involved would have been greater than the average person today could imagine.