From what my OR told me the final select takes place after you complete MEPS, background/security check and the PRT. The MEPS and the PRT results are reviewed in Tennessee to make sure you meet the standards. You shouldn't have any problems with this part because they tell you at MEPS if you are qualified. For what I was told if they qualify you at MEPS you are good to go. I have no idea how long the background check takes. I believe I have read on here, they only have to begin the check in order to get a FS letter. At this point your package is signed off and they approve you to swear in (final select letter). As far as I know you can swear in anytime after the FS letter has been signed.
The letter is mailed to your OR and he informs you when it comes in. I was told I would be getting a final select letter by Sept 1st, possibly sooner. I have read of several people on here waiting months to get FS letters. I asked my recruiter about this and he said they have made an effort to shorten the wait time for FS letters. He also said the longest wait he has seen was just over two months but that was a while back. Hope this helps.