It's the cops who charge people. It's the CPS who take it on from there. The CPS are not involved until there's been charges.

This is not quite the case..

Once the police have completed their investigations, they will refer the case to us for advice on how to proceed in all but the most minor and routine cases. We will then make a decision on whether a suspect should be charged, and what that charge should be.

Taken for the CPS web site.