For the Columbia-Juilliard program, you apply for the master’s program in your third or fourth year. Columbia-Juilliard Program | Columbia Undergraduate Admissions
You apply for participation in the program of lessons and performance (and practice rooms, master classes) at the outset, or while at Columbia and there are juries at the end of each year to determine if you continue.
For the fourth year, you can petition to continue one more year as is, or apply for the double degree.
Columbia has its own performance program Columbia University Music Performance Program and arts initiative Home - Arts Initiative at Columbia University.
For Harvard/NEC’s double degree BA/MM, you apply to both schools from high school. If you get into Harvard, you an also apply for the dual degree after the first year. You do three years at Harvard, but take lessons and participate in ensembles at NEC. Fourth year is mixed with more at NEC and 5th year is all NEC.
So the two programs are quite different. At Harvard you are admitted at the outset to the double degree program, but at Columbia, that comes later.
Harvard also has a double degree program with Berklee. And there are excellent musicians who only do music performance at Harvard while taking lessons either through Harvard or privately.
Tufts double degree is a BA/BS and BM, not an MM, with more involvement at NEC throughout. You apply to both separately. Unlike Harvard and Columbia, you cannot major in music at Tufts while doing the double degree.
All of these programs add cost in additional tuition but financial aid is available.