If you feel the combat in D&D is too long, why are you even playing D&D in the first place? I mean, that's kinda the whole point of the whole thing - gathering items, levelling your character, making them stronger for the fights ahead.
Making everyone have 1 HP and lowering the damage to 1 seems pointless and over-simplified to me. Consider this, it will make having different weapons redundant, wielding a dagger will be the same as wielding a two-handed greatsword. Also, how are you going to balance spells? By your logic, a level 1 magic missile will do as much damage as a 3rd level fireball. What's the point in leveling a character then or having different classes and play styles?
Seems to me like you're better off just switching the format. If you want to RP, you can remove most of the gameplay systems and just settle for a singe dice roll to determine success/failure. Alternatively, there are more narrative focused rulesets that have significantly faster combat than D&D, Savage Worlds and Edge of the Empire (my personal favourite) to name a few.