I made this house roll to reduce the frustration of bringing an opponent down to 1HP. This makes it more dramatic before death and adds minor realism to the game. This applies to both players and monsters.

Here's how it roughly works:

• A Creature is considered bleeding at 1 HP
• The only have 1 round left before they soft die. A bleeding creature can be attacked and hard killed
• Their movement is reduced to 10ft or half their normal speed if they already move less than 10ft
• This only applies to living creatures like beasts, humanoids, some fiends/celestials and aberrations/monstrosities.
• Their only attack action is to make 1 of their melee attacks with disadvantage.
• They have disadvantage on CON, DEX, STR, and CHA checks, and saves

Share any ideas in the comments if you think I could add/remove/change certain things.

So, I believe bleeding should be applied whenever a creature that is not a construct or undead takes piercing or slashing damage. A bleeding creature must make a CON saving throw at the start of their turn, taking 1d4 damage of the previously dealt damage (So slashing or piercing) on a success or 1d6 on a failure. AC would not effect the bleeding effect (Obviously) and a bleeding creature could use their action to make a Medicine ability check, bandaging themselves up on a success to cancel the bleeding effect. After the bleeding creature is reduced to half hit points, then a negative effect should be applied, such as exhaustion or an ability score penalty. Critical hits should dramatically affect the damage dealt by bleeding, like increasing the damage to 1d8 and 1d10. 

Magical bleeding should also be a thought. It would probably be harder to patch up than a normal wound, so the Medicine DC should be increased. 

Coughing/Vomiting blood could also be a possibility. If the bleeding creature fails the CON saving throw by rolling a 1, they could vomit/cough blood dealing around 1d12 bludgeoning damage. (Bludgeoning because coughing/vomiting feels like being physically harmed)  The damage would be instantaneous and would not have lasting effects, but the previous blood loss would still be in effect. 

A creature reduced below a certain number of hit points by blood loss would most likely fall unconscious. They would be unconscious until the lost blood is restored, via magical health restoration or natural regeneration. In real life, even if unconscious, your body will naturally repair itself, so while unconscious in this way would slowly repair their wounds. Depending on the amount damage dealt. So a hour or so would patch up most of the bleeding, stopping the effect. 

This is 100% a work in progress, so feel free to change anything. XD

The Coughing/Vomiting blood could be used for a new spell type similar to the Command spell and similar to Toge Inumaki's "Cursed Speech" from Jujutsu Kaisen.