I think the comparison of nail polishes to automotive paint is a bit wide of the mark. While there may be some similarities, the operating conditions are vastly different. That means that the formulation would be very different. Cars are exposed to a broader range of temperatures, ultraviolet light and sheet metal expands in heat much more than the human body does and doesn’t care about toxicity (in contrast to human tissue). I have no specific idea what causes your particular set of issues, but some searches turned up some interesting information…
https://www.thoughtco.com/nail-polish-chemistry-603996
https://rionnanailpolish.weebly.com/composition.html
I have never tried UV cure gel polishes, but I have tried gels with chemical hardeners. However, what turned me off was having to soak my nails in acetone. That didn’t seem like a very good idea at all.
Perhaps you might seek the advice and counsel of a dermatologist…