I received the NAMI Whammy whilst at OCS about 6 years ago; I doubt the process has changed much. Essentially, you'll meet with your Class Officer and he'll ask you if you want to stick around or not. If you choose to leave; they'll send you home with a medical discharge and a pat on the back. If you choose to stick around, they'll ask you what you want to do. If your Class Officer is squared away (mine wasn't), he'll have a good idea of what designators are actually available to you as they're all dependent upon the needs of the Navy and how badly each community needs people. After that, you'll sign some paperwork and do some quick re-detailing.
Here's the short version of my story (You can find the long version with some digging on the forum): I was NPQ'd with a week-and-a-half left in OCS. My Class Officer barely realized that I was about to be booted from the Navy and had an emergency meeting with me. In the meeting, he asked me if I wanted to stay in and commission (I did), and asked me what I wanted to do. I decided upon going EOD and he said he'd draw up the paperwork to get my designator changed. The next day I went and got a dive physical (passed it), and then went back to his office to sign my new contract. When I got there, he told me that the only thing available now was SWO and take it or leave it. I took it, then got my orders (I redesignated the same week that the Academy and NROTC were doing ship selection so the pickings were slim and I didn't even get to put in my preferences), and commissioned a few days later. So from NPQ to new orders, was about 1 week or 5 working days.