Before you jump to the waiver process, you need to understand DoDMERB
DoDMERB in essence is the medical clearinghouse for the military. It is not just DQ and Q, but also they may request a remedial.
Remedial just means we need more medical information before we decide if we will send to the commissioning source a Q or DQ.
The fact is you have a nut allergy. You must answer yes to that question. They can come back and ask for records from your allergist. Those records can move you into a Q, or into a DQ.
Once that decision has been made, it is the commissioning source to decide to waive the DQ. USMA may say yes, USNA may say no.
~ DoDMERB has no voice in this decision. It is the branches doctors that decide.
You cannot request a waiver. Commissioning sources request it.
~ AROTC may request the waiver because you were offered a scholarship. USMA may decide not to request it.
Good Luck.
My best advice is to get your medical records in order. Go to your allergist and see if they can write something in your favor. Your opinion that you can eat 15 nuts with no effect is not the same as your doctor saying it in medical terms. Take that with you for your intake exam with the DoDMERB doctor. Hope they will Q you, but expect that they will annotate to the Army's medical standards, you are not qualified without further investigation.
OBTW, as far as it goes with how long, it depends. It can be very fast if you are sent for the exam early on, but it could take 6-9 months if your waiver is in the bulk.