Answer:
I think you have done enough by flushing the product out of the cut and it would be good to wash the site every day and keep Neosporin on it continuously until it heals. I'm surprised that didn't burn a great deal when you initially got Nair in the cut! As far as long term affects to your system, there is not much it can do to harm you, once washed off. Hope you get better soon!
I had a cut from playing sports once on my leg and used Nair. I got a nasty infection and had to go on medicine. I would wait a couple more days than Consult your doctor. I am a medical student who just finished studying infections.
