When the particles that make up cosmic rays get boosted up to the speeds at which we see them, they're ionized. The electrons and nuclei are separated from the start.
Both types of particle are going to be releasing energy every time they accelerate (synchrotron radiation) and when they interact with photons (inverse Compton scattering). When it comes to inverse Compton scattering, the electrons scatter much more than the alpha particles - there's a $\frac{1}{m}$ factor in both the energy exchange and the angle involved.
What this results in is basically the electrons going in all directions, gradually losing energy. Meanwhile, the alpha particles continue on with less deflection and with most of their energy intact. The electron is the ping-pong ball to the alpha particle's bowling ball. (4 bowling balls, really, but you get the idea.)
We still get electrons from space, but more of them are coming in slow, which puts them out of the "cosmic ray" category.