As mentioned, your 10 business day wait starts the day after the LE agency with jurisdiction receives your information. For that to happen, you have to fill out the forms and sign them. Your FFL can't just submit an application based on a pervious transaction, it doesn't work that way.
Also, it is no longer a "private party sale" because your seller had an FFL ship it to your FFL. Your FFL will have to log it in from the out of state FFL, which means you'll also have to pay the Use Tax on it now. Had the seller shipped it himself directly to your FFL, then the FFL wouldn't have to collect the Use Tax. You are still responsible for paying the Use Tax in that situation, it's just not the FFL's responsibility to make it is actually paid.
5 days is a bit unreasonable to contact you, unless the package didn't contain any of your contact information, or the contact information was wrong. It doesn't sound like that should be the case here, but you never know.