I read somewhere that they had extended the NASS application period, but I am not sure that they extended the acceptance/notification period. In other words, you should hear by the end of the month. My DS's school is sending three kids to NASS this year, and one of them simply called admissions to see if he was in. That doesn't seem to be a problem, so you could give that a try.
I called admissions myself to ask the same question that you asked about SATs. They said to go ahead and submit them, but they were less than clear about whether it would affect Summer Seminar (as opposed to Official Candidate) consideration. They did tell me that it can take several weeks before the College Board's official score updates their candidate system. For that reason alone, if you decide to submit your official SAT at this late date and expect or hope that they will be considered for Summer Seminar, you should also send an email to admissions with an unofficial update. This was, again, per their instructions.
Not to ask the obvious question, but this all assumes that your scores in at least one category increased, correct? Don't forget that USNA superscores. You can submit the highest score on any section from any date.
The email address to use is:
webmail@usna.edu
This is the text of what my son sent, slightly redacted. Note that they don't ask for Writing, but his was strong so he sent it anyway.
"
Good afternoon. My name is Z, and my date of birth is MM/DD/YY. I recently applied for NASS and, since I completed my preliminary application, I have re-taken the SAT and improved my scores. Last night, I requested an updated official score report be sent from the College Board to the Naval Academy on my behalf, but I wanted to provide the following unofficial update.
Current best SAT Math: X
Current best SAT Reading: Y
Current best SAT Writing: Z
Thank you."