They've changed the requirements for this slightly, so take my experience with a grain of salt. But receiving loans that don't disperse until September should be okay. The UKBA knows US students receive loans that are sent directly to the university.
I had a Sallie Mae private loan and simply used an official letter from them detailing the amount of money that would be dispersed to my university, which date it would arrive, and which university it was going to. Sallie Mae do make a letter for visa use if you're applying for a visa, but it was absolute hell for me to get (I called them every day for a month and then contacted their lawyers to complain, and only then did they offer to overnight fedex it to me - at my own expense, of course). If you have been receiving letters from them by post, then you might be able to use those. If you're only getting the PDFs, you're going to need to call their terrible customer service centre and explain that they need to send you a letter detailing all of your loan information.
If you need any more info on SM private loans or want to see the letter I used for my visa application, just let me know.