My comments are based on a personal experience about 5 years ago, when my DL flight BOS-DCA was cancelled and I was stranded overnight at LGA after being rebooked BOS-LGA-DCA. Most of the terminals at LGA were closed at a certain point, but one remained open overnight, where 40-50 passengers spent the night inside, sitting on chairs or on the floor, but with access to restrooms. A shuttle bus between the terminals began to operate about 4 AM or so (sorry, but I can't remember the time more exactly), and I was able to go the terminal needed for my early flight. I had to wait for an hour or so before security opened.
I realize that things may have changed in the meantime, but I hope that these comments may be of some help.