I’m not sure which hypothetical schools your are looking at.
There are often two types of scholarships awarded to National Merit students - one might be for NMSF or NMF and another (generally a smaller amount) is awarded when a student becomes a NMS.
Very few schools ask to be indicated as the first choice before May 1. BU is one of them and has a March 1 deadline. This post from last year might help answer your question.