Where did you hear the undergrad experience was lacking? Harvard has lecture classes with grad student teaching fellows. It is often teaching fellows who interact with students, except in smaller classes and departments. As you move through the 4 years, this can change some, and you would have a thesis advisor, but overall, Harvard is a large school with TF’s (assume NU is too).
If you want interaction with faculty, you might prefer a liberal arts college, such as the “little Ivies” like Tufts, Amherst, Williams, Wesleyan and others. Also check Colleges that Change Lives website.
The best reason to apply to Harvard is their financial aid 
But some of those smaller schools have financial aid similar to Harvard’s (Amherst and Williams for instance).
I don’t see why you can’t apply early to Harvard and regular to NU, or apply to both in the regular application cycle. My main question to you is why, if you value close interaction with faculty, you wouldn’t be applying to smaller LAC’s with undergrad focus and more contact with professors.