Bizarrely enough, my guitar strings are nickel (the wound strings at least), where my bass strings are stainless.
I did notice when I was playing guitar more, that my finger tips were shedding like yours, especially when they were fresh. The nickels sounded great, where stainless didn't sound so good to my ears.
When I came back to bass, I started using nickels as well, but then I noticed that my fretting fingertips had more white/dead/shedding skin. I read something about it on here, and tried out a set of stainless. No more white/dead/shedding skin on my left and right fingers. I think it is more to do with all the strings being wound, and obviously a fair bit thicker than guitar strings.
Definitely worth trying a set of stainless next string change, if only to rule them out.