It seems that you are a “victim” of the misconception of potential energy that usually originated from the GCSE level.
At the GCSE level, students are usually asked to compute the gain in gravitational potential energy of an object with mass m that is raised from a height of h1 to h2 near the surface of the Earth, where h2 > h1. As a result, students have the tendency to think that potential energy belongs solely to a particular object. This is very unfortunate.
Consider another similar problem: A stone is held above the Earth’s surface and the stone is let go to fall under the influence of gravity. We often say the gravitational potential energy of the stone is converted to the kinetic energy of the stone at the GCSE level. Why?
When the stone is let go, both the stone and the Earth move toward each other, but the Earth’s motion is too small to detect (or Earth does not move significantly) due to the immense mass of the Earth relative to that of the stone. This means that the change in the separation of the stone-Earth system comes about largely due to the motion of the stone.
So, we often associate the gravitational potential energy of the stone-Earth system with the stone alone at the GCSE level.
In reality, gravitational potential energy is actually shared by both the stone and the Earth and does not belong to the stone alone.
Potential energy is always associated with a system of two or more interacting objects, so PE is a shared property among the two or more interacting objects. Thus, it is incorrect to call the electric potential energy of a nucleus or the potential energy of a particle.