By the use of mathematics, it's possible to come up with a description of how the electrons separate using conceptual entities that stays the same regardless of frame. This simplifies the discussion enormously - and simplifies calculations, as well. These conceptual entities, called four-vectors.
However that description involves four-vectors - four-accelerations and four-velocities - which is something you didn't want to hear about as I recall, because it involved math.
The math to describe how ordinary velocities and accelerations transform is more complicated than the math that describes four-velocities and four-accelerations transform, as the later are easier to mainpulate. I'd probably mess up if I tried to do the math for how ordinary velocites transform unless I used the techniques I'm used to, which involve four-vectors. Conceptually it's possible, I suppose, one starts out with how distances and time transform (but that still requires the Lorentz transform, which is math), then one needs to go through more mathematical manipulations paying careful attention to which frame one is into determine how velocities and accelerations transform. Finally, knowing how velocities and acceleration transform, one can discuss the dynamics, how the forces transform. But the math in this case tends to obscure the physics, because of the complexity of the transformations of ordinary velocities and accelerations as compared to their four-vector counterparts.
If we assume you still don't have the interest or background to talk about four-vectors, about all we can say is what we've said, which is that in some frames we describe the forces as having electric and magnetic components, and in other frames the magnetic components are zero. The components of the forces change when we change frames.
The simplest analogy I know of involves comparing changing reference frames to rotating maps. If we have a street-map, and we rotate it to crate a new map where north points in a different direction, in some conceptual sense the map is "the same map", even though we rotated it. But the components and the descriptions change. If one building was directly east of another building on the original map, it's no longer east on the new map, because the map has been rotated. The displacement has north-south and east-west components, on one map the north-south displacement is zero, on the other map it is not zero. But both maps describe the same territory.
