Ours took a lot longer than expected, not due to the US shippers but the people they contracted out to in the UK side. We had arranged it so that our shipment was supposed to arrive late October/early November (at latest). It didn't arrive until December. So, yeah, you should be prepared for delays even if you are told 6-8 weeks.
Anyway, I can give you some advice that didn't involve an international move. Several years ago, I moved to another city on short notice. I could only pack what I could fit in my car and had to wait until 4 months later to pick up the rest of my things. I look back at "roughing" it as being pretty fun. I didn't have any money to buy anything, and I knew no one. But when I did make friends, people were quick to help me out.
If I had to wait for my things here in London in unfurnished accommodation, I would look for the things I could use the easiest even after my things were here. If you're alone, you can probably make due with a lot less than you think you can. Air mattresses are an option, but an Ikea fold out couch or futon might be useful even after your things arrive, and probably has a better resale value:
http://www.ikea.com/gb/en/catalog/categories/rooms/living_room/10663/Speaking of Ikea, that's where I'd get some cheap eating and cookware as well.
I love those Ikea showrooms where they are like "This entire room furnished for under £300!" Sure, it's an extra expense, but I bet you could find use for everything you buy even after your things arrive.
ETA: I'd also sign up for Freecycle once you get here and put out a request for specific things I absolutely needed.