The interaction of two charged particles is not instantaneous, this would violate causality. In order to treat this system properly, you cannot ignore the energy and momentum carried by the electromagnetic field. All interactions are local between the charges and the fields and the charges do not interact directly with each other. If the charges are not moving uniformly (which they will not be if they are accelerated by each other's fields), then you cannot just take the "fields of a moving charge" that you will find in the standard textbook and compute the forces of those. You need to apply the Liénard-Wiechert potentials which will be coupled with the equations of motion for the particles themselves.
Edit: Also note that the A-level tag indicates that you have an understanding of the subject at the level of a graduate student or higher. As your questions suggests that you do not, I have changed the thread level to I.