I played in a 5e Zombie Apocalypse, wherein a Game of Thrones style Wall that kept back the undead fell, and most of civilization collapsed. We used a unique system to represent lasting wounds.
1. You didn't heal HP overnight, but you could still use Hit Dice, which did replenish overnight. Basically, a day or two of rest and your HP would be full.
2. Whenever you're at 0 HP, you aren't unconscious. However, you function as if you have two virtual levels of exhaustion.
3. Each time you drop to 0 HP or take damage while at 0 HP, you make a Con save (DC 10). If you fail, you gain a level of real exhaustion. These stack with the two virtual levels. If you reach 4 real levels and 2 virtual levels, you're unconscious and you must make a save each round to avoid gaining a 5th level, and then finally a 6th and fatal level.
4. Magical healing only let you regain HP, not recover exhaustion. Each week you rested, you recovered a level of exhaustion.
The effect in play was demonstrated in our first encounter, where my 2nd level dwarf fighter was grabbed, tripped, and gnawed on 5 zombies. I tried to fight them off, but I quickly was reduced to 0 HP, which earned me 1 real exhaustion level and 2 virtual. The cleric healed me, so the virtual levels went away, and I tried to keep fighting while the other PCs split their attention between the zombies trying to bite through my heavy armor and the ones that were trying to swarm them.
I ended up going on total defense, and my high Con let me succeed a lot of Con saves, so I survived three rounds being gnawed on by zombies before I finally reached 4 levels of exhaustion which knocked me out. The rest of the party narrowly managed to kill the zombies, then grabbed my limp body and ran onward. When we reached safety and they were able to make a Heal check to let me spend a hit die to heal, I awoke with, like, 6 hit points and 4 levels of exhaustion that forced the party to hide in ruins for an entire month to give me time to regain my strength.