If you apply online, it can be very quick, and it can actually be quicker than it appears to be. I submitted the FAFSA online on March 1, and have not yet received confirmation that it has been processed. But, yesterday (March 5) when I went online to check my status at the school to which I am applying, they had already received the info., and there was a weekend in there, meaning that it really just took a couple business days.
That said, it's not clear whether the info. is sent automatically to the schools you designate, or the schools must retrieve it in some way. The FAFSA Online website says, in response to the FAQ "How long does it take for my school to receive the FAFSA?,"
"Depending on when you file and how busy your school's financial aid office is, it could take anywhere from a couple of weeks to several months. You'll need to inquire at your school's financial aid office for status updates about your FAFSA information after you've filed it."
Now, when the school receives the FAFSA info. and when they process it on their own end to arrive at a financial aid award are two totally different things. I've been using the FAFSA (first paper, then electronic) since I started undergrad in 1995. Sometimes a financial aid award comes with your offer of admission, sometimes it comes afterward (which obviously can make it difficult to make a decision about whether to accept the offer of admission.) And as the previous poster said, sometimes the FAFSA info. is the only info. the school looks at, other times they have their own (sometimes more detailed) application.
Just to note to anyone who hasn't applied yet and wonders whether they should do their taxes first.... of course you can apply with estimated info. and update it later. BUT, I had the smoothest application ever this year, because my return had already been submitted and processed. FAFSA online was able to import the income info. directly from the IRS so I didn't even have to find my copy of the return and answer the questions myself.
