Sometimes deserters came into Federal lines in small groups, and sometimes alone. MRB came into Nashville two or three weeks after the Nashville Campaign ended. At this time there were no organized Confederate forces anywhere in west or middle Tennessee.
We don't know where he was when he left his command.
If he was with the 37th Mississippi Regt., MRB probably deserted while the regiment was just south of Nashville. That means he remained in the area, eating acorns and hiding from Federal soldiers for a month or so. Why not go ahead and turn himself in? I really don't inderstand this course of action.
If he was with the 54th Alabama Regt., MRB probably deserted while the regiment was stationed on Mobile Bay. That would explain what he was doing for the next few weeks, but why not go ahead and turn himself in at Pensacola? I don't inderstand this course of action either.
If seriously wounded at Resaca, he could have been sent home on furlough or to a Confederate military hospital. If so, when released six months later, he may have decided not to return to the army. For instance, if he was in a Montgomery hospital, he could have made it to the nearest Federal post, Chattanooga or Decatur AL, and gotten transportation to Nashville from there. In that case he almost certainly came to Nashville to take the oath among a group of fellow deserters.