The time it would take would vary greatly depending on external factors. Here are just a few examples.
What condition is the individual in?
Age and body size plays a factor in axolotl regeneration, so it's reasonable to assume it would also play a part here. Someone young and in good shape would likely regenerate faster than someone older or with a worse physique.
What sort of injury is this?
Healing a cut or stab wound would obviously take less time than regenerating an entire appendage, but if the damage is extensive or the wound has caused mutilation, healing would take longer. In some cases, you might even find that mutilated tissue damage takes longer to repair than it would to replace.
What sort of healthcare situation is the human in?
There are many things that could shorten the necessary time to heal, such as:
- Remaining in a sterile environment (i.e. hospital or laboratory).
- Maintaining proper general health, such as staying hydrated, maintaining a nutrient-rich diet, and staying as active as the injury allows without further damage to the surrounding area.
- Getting proper time to rest.
With this in mind, injuries sustained while in a combat-heavy zone or wilderness region, with the injured party eating field rations and getting 3 hours of sleep every night, would spend a lot longer healing than someone with access to proper medical treatment.
What do you consider "healed"?
Let's say you end up having a hand cut off. What would you consider "healed" to be in this scenario?
- The appendage re-appears, but is unusable (no feeling, no motor control, etc.)
- The appendage has feeling, but no motor control.
- The appendage has motor control, but no feeling.
- The appendage has both motor control and feeling, but is sluggish and the injured party still suffers pain when using it.
- The appendage is perfectly healed, as if no injury ever occurred.
Depending on how the regeneration works, the patient may go through some or all of these stages while recovering.
With all that said, let's assume that you have a young, perfectly healthy patient with no pre-existing conditions to affect recovery rate. They're in a hospital setting, getting plenty of nutrition and rest, and all they're recovering from is a deep cut across the inner elbow. Severed muscle and nerve endings, so the arm is useless from the elbow down, but they're not in need of a completely new limb, just some regenerated connection.
Given all that, a juvenile axolotl takes 40-50 days to regenerate an entire limb. Now, terrestrial salamanders take much longer than axolotl do, but that gets into an entire tangential discussion, so we'll throw that out. We're just regenerating a fraction of that, I would say it might be reasonable to estimate that it will take 1/5 of that time, say...8-10 days. Of course, you could argue that humans are a lot bigger than axolotl, but we also have a lot more active healing systems as well, so our size could conceivably balance things out.
Note: A lot of this is conjecture based on cursory research. I am not a geneticist.