Addition is faster then subtraction, so combat is not slowed down. The only potential problem with this system is that somebody could make an error and not notice they went over their Hit Points- but unlike a subtracting HP total, an earlier mistake won't mess you up for several rounds in advance. I'd write each injury individually, which gives you a nice little history of your misadventures, but you could run several running totals for each damage type if you like. You can also keep a total Hit Point counter to save time and easily tell when the next hit kills you.
In many ways, this system actually speeds up combat. Now you can tell instantly what wounds are killing you. It also helps for special powers or abilities- the character who is resistant to bludgeoning damage just writes half the value down on their sheet, instead of doing subtraction twice. The werewolf can regenerate 1d6 hit points every round- except the silver wounds, which are now separate from their hit point total so you won't get confused.
Extra – What about Health Potions?
All health potions are “Potions of Healing”- no minor or major or grand or whatever. They restore all your damage, instantly. You can't use other healing methods to heal in combat. This may be too strong- but if you make Potions of Healing rarer it would be have the same impact. You can also make health potions that heal specific types of damage, or an amount of damage, if you want to fiddle with rules more. Personally I like the idea of specific cures that can be used in combat- like a rubber-blood potion that seals up all your bleeding wounds, or a potion of liquid warmth which cures your cold damage and stuff. Doing individual amounts like traditional d8 healing potions or healing spells would mean that different damage types would have to have a sort of set order, or letting the players pick what they are healing. I feel like this works too- drink the potion to cure internal damages, pour on the skin to cure cuts and bruises, etc. So it's not a huge stretch to let a healing potion heal whatever damage types you want.
I also wrote this concept up in this alchemy ruleset- but I also like the idea of cheap or weaker health potions with a short shelf life- tonics. The idea if players can make or produce them on an adventure, to fit with the theme of being a magical healer, and to give magical healing for in-combat stuff, but they expire and can't be saved adventure to adventure to prevent the problem of stockpiling too many (or by contrast- never using them because they want to save them until later).