Does it take a long time for vats-related nerve damage to heal?

It can. Although vats incisions are small, they still require cutting through skin, tissue and possibly nerves. This may lead to chronic numbness or even pain. Nerves grow back 1mm per day.
Yes. If the nerve damage is to skin and muscle the nerve may take a long time to regrow. Sometimes numbeness or neuropathic (nerve related) pain may take 6-12 months to go away completely. The good news is that there are a lot of new medications to help with pain control after vats.