I am also 39 weeks tomorrow. I've been actually scheduled for a c-section for 26.1. because I wouldn't dilate not even to 1cm (basically no progress since week 34). I went in today to do the pre-surgery blood works, ekg, etc. And my dr thought to check my cervix just in case. And guess what, I am dilated 2cm and thinned-out (don't remember how much) but enough to have my c-section cancelled. The dr was also surprised because she really didn't expect this. Besides, when getting home I noticed some brownish discharge in my panties which I think might be the mucus plug or a part of it. So, baby come on out - I am ready for you :)
I was induced with my first and I was at 2cm at 40 weeks and I dilated quickly. Everyone is different though...