Someone from Florida will have to answer your specific questions. I've had CCW permits in VA, NC, and AZ, all of which took about 90 days for the original permit. Less time for renewal in AZ (about 2 weeks, but 90 days for renewal also in VA and NC. In VA for a non-resident permit renewal, it isn't really a renewal, you have to submit everything again as if you were applying for a first time permit.
You obviously have to pass the NICS check to get the CCW, but at least here in NC now the Sheriff will also do a more in depth background check witch includes checking your mental heath history. I have no idea what Florida will do, but since it isn't a 'constitutional carry' state (meaning if you can legally own a firearm you can carry it concealed) I would expect that the NICS check done when you bought the gun alone won't be sufficient, there will likely be a more in depth check. Again, it would be best if someone from your state with knowledge of your state laws would answer.