Also, my apologies to anyone who had questions I didn’t get to that appear to have been answered by others! I’ve been extremely busy in the past few weeks and haven’t been on here. However, if anyone has any additional questions, I would be happy to answer 
If I may add a couple of points for @WorriedNow101 :
I don’t think admissions as a whole is all that biased against off-campus majors. When my d applied, she indicated interest ONLY off-campus majors (CogSci and CompSci), and was still accepted with merit aid. Individual staff members may be less informed and encouraging than others about non-Scripps majors, but that shouldn’t be a deterrent. If you want to major through the consortium, you absolutely can. (If there’s bias at all, I think it’s less about off-campus majors per se than about students seeing Scripps only as a back door into another colleges’ programs. But you clearly appreciate Scripps in its own right as well, so that concern does not apply to you.)
Further, the Mudd CS department very clearly states their policy that off-campus majors have equal priority to Mudd students in accessing Mudd CS classes. (And as @adaserk said, you can declare at any time.) Mudd CS will not treat you as a second class citizen in any way. The only restriction is that you cannot minor in CS through Mudd - if you decide to minor in CS, it has to be through Pomona. You can major through either, and Pomona has an excellent CS department too, although Mudd has a bigger “name” in the field. You can take courses from both, regardless of where you major in based. The only thing you can’t do is mix courses in the intro sequence between the two schools. The Pomona and Mudd CS departments organize their three-course intro sequences differently, so once you start one sequence, you have to finish it before you’ll be free to pick and choose classes from both programs.
You may need to be okay with starting the CS sequence in your first-year spring. In your initial semester, you will have two required classes (Core and Writing), and you’ll need a math class if you plan to do CS. Your fourth class is more likely to be a GE requirement (probably a lab science or social science). Intro CS is a required fall class for Mudd first-years, whereas non-Mudd students more often take it in the spring. It’s not 100% impossible to get into it in the fall, but it’s not a sure bet, and workload-wise there are sound arguments for getting acclimated to the Scripps core curriculum first. If you’re totally determined to hit the ground running in CS, you can probably get into an intro class (may be easier at Pomona than Mudd?) but don’t be surprised if it’s difficult and/or if your advisor discourages it. There’s still plenty of time to complete the major if you start the sequence in the spring.
Hope that helps!
Side note: I really hate how CC doesn’t let you fix typos after the first 15 minutes. Please disregard the various literacy fails above.