Nickel allergy affects abut 10% of the population to some extent. Once it gets severe enough to notice it will probably go on getting worse for the rest of your life, and there is nothing you can do about it except avoiding exposure.

Nickel is by far the most likely cause of the problem.

Googling "nickel-free guitar strings" gives a few hints. Though it is always possible the supplier of such might be lying (most "nickel-free" jewellery isn't).