It's not too late, as long as you keep on top of things. The private loan (I used IEFC/TERI, so that's my experience) tends to go much faster than the Stafford because they don't have to deal with the guarantor crap that tends to slow down federal loans. All the school has to do is sign the certification form and then they schedule disbursement within 48 hours (and my school was given the option of faxing the form back, which obviously saves loads of time- the stafford forms HAD to be mailed back, however). However, my private loan was actually delayed for almost a month because I didn't realize that the school hadn't received the certification form that IEFC sent (probably due to an outdated fax number, I'm guessing.) So as soon as they send your certification form make sure the school has it and if not ask them to resend it to a fax number you specify.
The Stafford loan will be trickier to pull off, but as long as you have a cooperative university (essential- make sure you know the name, email address, and phone number of the person who will be signing your papers and hound them to do it in a timely fashion) and you are willing to follow up every step of the way it should be okay. The biggest thing for the Stafford loans is to be on top of the guarantee process (after the school mails your form back to the lender.) This is what holds up the loans of many international students because the guarantor agency (completely separate from the lender, part of the government) has to get in touch with your loan contact (mine did it via email) and they don't always have the correct information. So, as soon as you know the paperwork has been received by the lender and sent to the guarantor, make sure you ask them exactly who your guarantor is, and then look the guarantor's info up on the web and
call them yourself. It was communicating with the guarantor and making sure they had all the info they needed (and then making sure my loan contact at the uni responded to their email) that got my guarantee pushed through in a matter of days instead of weeks. Once you've got your approval- if you're really pressed for time and you can't wait the few days for the actual letter to come in the mail for your visa application, get your lender to fax the letter to you and use that instead.
You'll be cutting it close for sure (is there any way you can apply for your visa in person? That will make things much easier) but I still think it's doable. The key is to call, call, and keep calling- make sure you know what's going on at every step of the way, and make sure things get signed, etc as quickly as they can. Remember, no one really cares about getting your loan done quickly except you so if you're not on top of it it will be easy for your application to fall through the cracks. Good luck.
edited to add: I was very pleased with my IEFC experience for my private loan. They were always very helpful and responsive when I had questions, and my uni was used to dealing with them (and they have a London office so it's easy for UK unis to get in contact with them themselves.) I loved being able to apply online for my loan (something you can't do for overseas schools through the regular TERI site) along with checking my loan status online, which was a big sanity saver at times. Customer service was amazingly better than my Stafford loan lender (Bank of America) but I actually didn't ever spend that much time on the phone with them because the hassle was minimal. IEFC definitely gets a thumbs-up from me
