I can't speak to how long it might take to heal, I feel like this might vary based on the age and general health of the dog.
However it wouldn't surprise me at all that a cut on the pad would cause the limp. One of my dogs will start to limp if he even gets something stuck on his foot or leg furnishings (burr, stick, acorn). Obviously, in the case of my dog, it's a bit of a dramatic response, but for you dog, the paw probably hurts, especially when she puts weight on it, so she's favoring that leg.
However it wouldn't surprise me at all that a cut on the pad would cause the limp. One of my dogs will start to limp if he even gets something stuck on his foot or leg furnishings (burr, stick, acorn). Obviously, in the case of my dog, it's a bit of a dramatic response, but for you dog, the paw probably hurts, especially when she puts weight on it, so she's favoring that leg.