The understanding of particle transport mechanisms in biological and synthetic hydrogels is crucial for the development of advanced drug delivery methods. We propose a simple model for the diffusion of charged nanoparticles in cross-linked, charged hydrogels based on a cubic periodic environment and an electrostatic interaction potential of varying range and strength, encompassing attractive and repulsive scenarios. The long-time diffusive properties are investigated by use of Brownian dynamics simulations and analytical methods. A number of experimentally observed phenomena attributed to nonsteric interactions between hydrogel polymers and diffusing particle are naturally reproduced by our model. Charged particles diffuse slower than uncharged particles, regardless of the sign of the surface charge, but with a stronger hindrance effect for attractive electrostatic interactions. This is explained in terms of charged particles sticking to the polymer network in regions of strong opposite charge and their exclusion from similarly charged regions. In the case of attractive interactions between hydrogel polymers and the diffusing particle, smaller charged particles diffuse slower than larger ones. This stands in contrast to a size filtering scenario but is in agreement with experimental findings. In the case of repulsive interactions, a range of differently sized particles diffuse equally fast. We compare our model predictions with published experiments on charged particle diffusion in hydrogels and confirm that electrostatic interactions are a key factor influencing the diffusivity of charged nanoparticles and that oppositely charged gels are much more effective in slowing down a charged particle than similarly charged gels.