As long as the cooked chicken smells okay and has no mold on it then it's likely safe to eat the cooked chicken after being left out all night.
I've left some cooked chicken out overnight and then reheated the chicken the next morning and ate it and had no health issues or got sick from it.
It's best to try to remember to refrigerate any leftover cooked chicken you have to prevent the potential of bacteria growing on the chicken.
But if the chicken has been cooked it's less likely to grow bacteria and go bad when left out overnight unlike raw chicken that grows bacteria and spoils rapidly if not refrigerated or frozen.
If you've left out raw chicken overnight you should throw it out and not cook it or eat it because it could make you sick.
But although it's not recommended I have eaten cooked chicken that I left out overnight without getting sick.