we all know how long it takes from an egg being laid, to the egg hatching and then the brood emerging, so what exactly is your question trying to ascertain?
I've had queens emerging, mating within days and laying straight after, introduced queens being released by the bees and starting to lay within 24 hours or so of being introduced.
On the other hand (apart from there being five fingers and a thumb) I've had queens taking over a month or so from emergence before I've seen evidence of laying, and introduced queens the same.