Don't worry too much about what degree you have when it comes to EE vs. CompE (and even vs. CS to some extent). It's far more important that you take classes related to what you want to specialize in. During my interviews, my interviewers didn't care what my actual degree was (it was EE), only how my project and coursework related to the position I was interviewing for. In most programs, EE and CompE overlap quite a bit. You could even have 2 students take the exact same classes yet end up with different degrees - 1 EE and one CompE, depending on how they structured their electives.
Regarding CPU design - your position in the CPU design process as a new hire will depend largely on your relevant coursework and project experience coming in. EE, CompE, and CS degree holders are all involved in chip design.