This has not happened to me, but, thinking about what happens to moms at the start of nursing a newborn, when they get abrasions from poor latch -- I think this is pretty much the same thing.
When I nursed my newborn, I struggled so much with poor latch that I constantly got abrasions that would barely heal and then re-open with feeding. It was so painful, it felt like torture. I was told by many to air out my nipples and that did not seem to help very much, especially with the pain. I applied milk to them, walked around bra-less and sometimes totally topless (not by choice, my nipples were so painful nothing could touch them), the whole bit. Then, I got the recommendation to use moisture to allow more effective wound healing and that healed me up in a day and a half. Here is what I did -- and I warn you, it's a little weird...
I got a box of Saran wrap that I used solely for this purpose. After breastfeeding, I dried my nipple with a gauze and then applied a very thin layer of bactroban on it (for MRSA protection, and available by prescription only; your OB or family doctor or even your pediatrician can help you out here). Then, I applied an approximately 5 x 5 inch square of saran wrap over my nipple and I put my bra on. I carried clean 5 x 5 square wraps in a clean place, and every time I fed, I repeated the entire process (gently dry the nipple, THIN layer of bactroban, not a glob of it, and finally a NEW 5 x 5 wrap, no re-using!).
I went from nipples that were tough, scabby and so sensitive that I could not even wear a bra (and this lasted for weeks!) to nearly totally healed in only a 1 1/2 days and I was able to feed with much less pain. Truly, it worked wonders; so well that I am still baffled as to why this is not recommended more often.
I strongly recommend this for your cuts. I was told by many that I could increase my risk of thrush; eventually this did occur after several weeks, since I suffered with poor latch for weeks and weeks, but this is not the same problem here. When Saran wrap is used for brief periods of time there should not be a problem. As a preventive measure, if you want to be extra careful you can take probiotics and attempt to rinse, not just dry, the breast tissue, with a little clean water on a swab or cotton ball to completely get rid of saliva that may be harboring thrush. To further speed up healing, it is perfectly fine to stop feeding from that side for a day or so; instead, to avoid further problems I would recommend pumping 2-3 times that day on that side so that you are not uncomfortably engorged. After about a day, you should feel well enough to go back to breastfeeding as usual, keeping up with the Saran wrap method as needed until you are totally healed.
In the same way that a bandaid allows for wound healing by covering the wound, therefore retaining some moisture...this is the same idea. Also, Medela and other companies make those gel-pad thingys for sore nipples. Again, the same idea, but much cheaper and you get days of treatment from a simple box of Saran wrap (as opposed to only 24 hours with those gel thingys, which should not be reused).