The whole process differs for everyone. But since the end of the FY is coming, it shouldnt be any later than late august, since your not officially in the program until you swear in, which has to be done be the end of the FY.
After you submit your application, its sent to a processor to make sure there are no errors on it, and it gets sent back to your OSO for you to sign. You have to take the ASTB before anything else at this point. If you get good scores on the ASTB, then your whole package goes up to the board. If you are pro'recd then you go and do the physical, and PRT.