About the fastest replication time I'm aware of is a single celled bacteria like E. coli at around 25 minutes. But more complex differentiated cells require more time because there are more organelles to replicate and the DNA replication process is slower due to more DNA and a higher level of damage detection/repair. I think most human cells take 12-24 hours to replicate, and that time is for fast replicators like skin basal cells.
The problem with "natural" body repair is that it is unintelligent. A liver cell isn't really aware that half of the entire liver just got blasted into mush. It just responds to changing chemical gradients, a variety of feedback stimulators and inhibitors, and increased cellular demand (since there are fewer liver cells now). So this takes time and liver cells are pretty specialized, taking longer to divide than a comparatively simple squamous cell. A lot of repair isn't returning the body to an uninjured state, it is laying down scar tissue that keeps the body working, sorta. Maybe down the road some of that scar tissue and granulation tissue is replaced by healthy functioning tissue, but there is still evidence of injury.
So you need external intervention. Nanites might provide this, as they can be directed to a specific organ and set up an extracellular matrix that will promote a more rapid growth of the "good" cells and less growth of stabilizing, but function limiting, scar tissue. This is still slow compared to, say, Wolverine's healing factor, but would be much faster than natural healing (perhaps days/weeks instead of months).
Another possibility would be pluripotent stem cells that could quickly be differentiated into specialized cells. Probably still need nanites to lay down a framework to hold everything in place and provide the initial chemical prompting to differentiate, but if you could fill the spaces with stem cells they might differentiate faster than new cells could grow. So you would need an external source of stem cells or at least a large reservoir of them within the body. This still wouldn't match Wolverine, but could get significant healing down to days. Nanites alone might be able to seal small skin lacerations and stop capillary blood vessel hemorrhage within seconds, but I doubt they could stop an arterial bleed or hold a fractured bone together while the person continues to fight with the limb.
So a major injury is still gonna be a major injury, but provided you could get some rest, have a good supply of nanites and stem cells, and have a LOT of water, fats, carbs, amino acids (the raw material for cellular division and differentiation) you could start healing immediately and be well on the way to recovery in just a few days. Control of damaging inflammation, good antibiotic meds to stop infection, pain control, and nanite repair of nerves will also be required; healing isn't just about growing tissue.
The most rapid form of repair would be replacement. Got a busted arm? Replace the arm. Obviously to have easily replaceable organic body parts would require significant rewriting of human anatomy (to have "quick attach" nerves, tendons, and blood vessels) otherwise you have to use cybernetic prosthetics.
You can substantially boost performance by adding in some redundancy to the human body. A second heart, for example, would help, as would a better blood reservoir (loss of blood pressure due to hemorrhage or cardiac injury are major causes of rapid death). You basically want to buy time for the nanites and stem cells to get to work while still allowing for at least some mobility and presumably combat functionality. The ability to ratchet down on inflammatory and pain responses would also improve functioning, at least for the short term. Getting a burn or exposed to chemicals often isn't so crippling, but the body's response can be (since it wants you to sit down and heal instead of get additional exposure). We evolved without hospitals and critical care, a lot of our bodies injury responses don't take that into account.
One thing you want to be REAL careful of is cranking up a cells ability to replicate without having SUBSTANTIAL control mechanisms. Otherwise everyone will be riddled with cancer. Our immune system already deals with rogue replicating cells on a daily basis, if you suppress the immune system but ramp up cellular replication then cancer is gonna be a big problem.
