Do people around him have to realise he died of his old wound?
Neither is implausible, but it's much easier to see how it happened without anyone knowing. Old, improperly healed wounds can kill you in all manner of roundabout ways. They can harbour blood clots that come loose and give you a blockage. They can be home to weakened and overstrained blood vessels that burst one day without warning. they can have bits of projectiles wandering around your tissuees until they find something vital, or rub against something important until it gives. All of that would not necessarily be directly connected with the wound by contemporary medical knowledge, but quite deadly. A 'sudden stroke' or 'fatal seizure', in the parlance of the day.
A wound can also reinfect, either because it failed to properly heal and was at some point exposed to pathogens, or because an infection was never quite subdued and a temporary weakening of the immune system allow it to break out. This might well be recognised as emanating from the old wound, a case of 'blood poisoining' or 'septicaemia'. Some wounds also create side effects making people vulnerable to infection despite themselves healing fully, through scar tissue that ruibs them or tretches tender skin. This does not have to be a serious wound or produce any great handicap in daily life. There is no known remedy for infections at the time, whether or not it kills you is down to luck and strength.
A man getting shot in the lower leg at Gettysburg, with no bone damagem, and even gets to keep his foot by some miracle (maybe not seeing a surgeon immediately) with some loss of mobility would still be capable of working, functioning in society, and conjugal relations, but might easily one day drop dead because a blood clot from an improperly healed vessel in his calf that had been innofensively sitting there for years dislodged and travelled to the brain, or because that persistent sore spot where the bullet nicked him suddenly swells up and goes putrid.