For me, the adrenaline or whatever took over, and sleep was the furthest thing from my mind during labor. I hadn't been sleeping much in the last trimester (mostly working long hours, then normal 3rd tri sleep disturbances like pee breaks and muscle cramps). On the day labor started (a Monday), I woke up at about 8:00 am, and DD was born at 6:45pm on Tuesday, and I didn't sleep at all during that time.
I was getting ready for work on Monday when light contractions started, so I stayed home and was too excited/nervous to sleep. Contractions got a bit more serious by about 10pm Monday night, and my doula told me to try to get some sleep then, but the contractions were too sharp so I couldn't sleep. I labored at home until 3am Tuesday morning, then went into the hospital when the contractions felt super intense and serious--but I was only about 3cm dilated then. Contractions got more and more intense until transition ended at about 3 pm--no way I was sleeping through any of that, although I did go into kind of a transe-like state in between contractions that I think helped me recharge a bit. And from about 3-6:45pm Tuesday, I was pushing. I never had an epidural or any such medication.
So I went almost 35 hours without sleep up to that point, then stayed up that evening until at LEAST 10pm or midnight, so more like 38-40 hours without sleep. I was fairly delirious and wrecked by the time DD was born, but at no point did I feel sleepy. It certainly would've been more pleasant and maybe faster/easier with more rest, but I wouldn't stress too much about it. Just try to rest as much as you can, and I'm sure you'll be fine in any event.
Mom to sweet DD 8/08, and adorable DS 10/20/10