What do you mean by "at first"? The protons are pushing each other away via that Coulomb repulsion while they're all in the pre-fission nucleus, but at that time it's not enough to counteract the strong force (https://en.wikipedia.org/wiki/Strong_interaction) holding the nucleus together. Once that once nucleus has been split into parts that can move freely, the daughter nuclei are both accelerated by an electromagnetic force.

Well, yeah, once the distance gets out of the well of the strong force, the nuclei are at the top of a big electrical force gradient that gets converted to kinetic energy. With proper ionization of the daughter atoms that kinetic energy could potentially be converted back to electric, well, potential.