For What It's Worth:
Computer scientists/engineers can work their way in to most all of the engineering disciplines; the others, not as much. I speak from experience; in my current job, I get to work with everything from rocket fuel to concrete, and my degrees are all computer science. You learn a few things in CS that generally apply, like formal logic and complex system organization. With either of the two majors you identify, look into the internship programs with the big aerospace contractors: LockMart, Boeing, Northrop Grumman, etc. You'll get an ear/eyeful and maybe even your security clearance in a summer's soiree...
But, sometimes I wish I'd done the machinist thing...