I don't have a Cartier watch however, there was one point in which I did think that I had an allergy to one of my stainless steel watches.
But I figured out what the problem was. It wasn't an allergy.
I was wearing the watch very tightly, as in, it was always touching my skin without any room for the skin to breath. My sweat would build up under the watch and then my skin would get red and irritated and start to peel.
Give your skin a bit of break, let it breath and see how it is. Maybe wear your watch on the other hand for the time being or wear it more loosely.
Do you have any other stainless steel or gold jewelry? If you don't have an allergic reaction from those, it may not be an allergy.
Good luck!