Response time depends entirely upon the individual faction. While most factions will get back to you within 24 hours, some factions (most notably the Rebel Alliance) have policies of performing background checks, or other procedures, before allowing the new applicant in.
If you are applying to the RA, I suggest contacting David Kellar by Darkness Message, and he can probably give you some advice of what to expect, what to do while waiting, if the process can be sped up, and so on. For any other faction, you might want to try visiting their website, post on their forums, see if someone can give you some advice, or look into it for you.
Or you can simply reconsider if you really want to join that particular faction, and start looking at others.
____________"May the Grace of Ara go with you, and His Vengeance be wrought upon your enemies."
Only fools and children dream of heroes.