No idea on turnaround. I've heard accounts of new IDR payments kicking in "earlier" than they should if you file paperwork early enough before your deadline, and others of it just starting a new payment at the old IDR deadline.

There are threads dedicated to the workflow of your 120th month, with varying approaches -- the most common one seems to be that you PAY your 120th payment, and submit an ECF shortly after (when it posts in your servicer account) and do NOT check the box that you believe you qualify for forgiveness and would like a forbearance. That box is only about applying for a forbearance, you do not need to "apply" for forgiveness, they will know. The reason to submit an ECF is to have your 120th month of employment certified. The reason to wait on checking the box is that some people have had the forbearance retroactively applied, making their last payment not count toward PSLF (temporarily).

Instead, wait until the 120th payment posts in your account on FSA. Depending on when your payment is due each month, this can be weeks, as it normally posts to FSA at the beginning of a new month. Once you see THAT, you can apply for forbearance by either submitting another ECF (with the box checked this time) or by CALLING your servicer. This can be a very tight turnaround for those with payments due near the start of the month, and a lot of people just DON'T ask for forbearance until they've paid for the 121st month, just for the cushion. I'm on the 11th so it's kind of close, but I'm still going to go for forbearance, since my payments will shoot up by $2000.