I went through this end of August/early September as I wasn't paying attention to the calendar and time just kinda passed. So I had to renew with 3 weeks or so before our trip. I thought I was on top of things but then I forgot to include my original passport in the first package so I had to book a new courier again.
Ill tell you straight up that the biggest problem is the numbskull courier company they use. Seriously - solid hard core idiots work for DX. Poor communication options, poor communication when I could get through, and they charge you for calling in. The US embassy whipped my passport home and through the system in about 8 days, about 8 other days were spent trying to get hold of the couriers to have a question answered, and then, when it came back, trying to find where the hell my passport actually was as I needed it ASAP. We had issues with a courier not being able to a) find the delivery building (in central London? Really? Yeah right) and then b) saying they attempted delivery when it was to an office post room which was manned when they said they were there. Just on and on.
To be fair when I finally called them up and demanded where my passport was and that I needed it to fly in three days (more like six but I wasn't going to give them a chance) it magically appeared and was delivered next morning at 11. I vaguely seem to recall at one point emailing the embassy and maybe they got on the courier company too but yeah it finally got to me with oh, 4 or 5 days to spare. Just a bit too tight for peace of mind!
So, the US embassy will get it through (although xmas...). If you have a straight forward delivery you ought to be ok, but if you have any sort of a complication like my two packages/two tracking numbers issue then the wheels are probably gonna come off the bus (just be prepared for some suckitude). You ought to be ok for three weeks though, even with Christmas in there. Some people have had much better experiences within three weeks, so let's hope you are included in the latter group!