This has been answered several times recently, but of course looking for the last 15 minutes I couldn't find any of the threads!
Basically, an LOA means that if you a nomination - ANY nomination - that you will receive an appointment (pending medical approval of course). That's why so many people are so excited to get one - it's kind of a way of doing the routine backwards! As long as you make the list of 10 on your MOC (Member of Congress)'s slate, with an LOA, you will get in.
However, the more normal way works in reverse. The congressman submits his 10 names, and if it's a competitive slate (ie. he just lists the 10 names and lets the academy choose who they believe is best), then it takes a long time because the academy CAN'T decide which of those 10 is the best until all 10 applications are complete and they compare them to one another. Since the applications aren't due until late Feb/beginning of March, even if you get a nomination and have everything in your application done already now, you won't get the appointment until all 9 of the other people on that slate are also done. And then you have to be the best of the 10, whereas with the LOA, just any of the 10.
Now if the MOC ranks their slate in order, and/or declares a "Principal" nomination (ie chooses someone as number 1), then by law, if that person is qualified, the academy has to offer them an appointment. It won't go on to anyone else unless they decline it (or aren't qualified). So you'd need to find out how your MOC ranks. Many do competitive. A lot of times if they do principal and you ARE the principal, they will tell you, but may not otherwise.
So in answer to your question, without an LOA or a principal nomination, the odds are that you won't hear anything until March (unless everyone on your slate is on the ball!). And even then, unless you're number one in the academy's eyes (and thus offered the appointment at that point), you still may be waiting while they see if #1 turns it down, or if you can get a slot from the wait list or somewhere else. So it's unlikely that you'd get a "no" response (rejection letter) anytime soon unless it was obvious that you were clearly unqualified (not likely if you got a nom anyways!).
Good luck!