Iris' cord stump fell off after about 10 days. We did have one Wee Notions nappy that snapped down, but the rest of the time she was in size 1 Tots Bots (similar to LL, I think). They gave her an enourmous bee-like behind, but they did the job.
She was an ounce under 9lbs at birth, but she lost a lot of weight and was closer to 8lbs at one point, so I think you should be able to use your LL straight away. Don't forget that you can always nippa down the LL if you think they are too big - should help them fit a little better. I also found that some of the BTP nappies I got were a good fit right from the start (and still are): Bumgenius may not be the prettiest nappies, but they are very functional and good even with explosive BF poo.
I did a review of our nappy stash in the newborn review section, if that helps.