The question of when the stories of the Nights were "officially" told by "Shahrazad" and who she them to who seems contentious and different in most every version of the Nights and is an interesting aside that I notice as I'm studying them and perhaps will come back to if time warrants in the future.
In the last section of Burton's Supplemental Nights vol. 1 which mostly seems to contain stories from the Breslau edition of the Nights Burton writes in a footnote:
"1 The Bresl. Edit. (xi. 318-21) seems to assume that the tales were told in the early night before the royal pair slept. This is no improvement; we prefer to think that the time was before peep of day when Easterns usually awake and have nothing to do till the dawn-prayer" (250).