Besides the nest box, it's best that you get all the hand feeding supplies in case you will need to raise the chicks.
Since the old seed cup turned into nest hasn't been used for feeding purposes for a good while, you can remove it and place the nest box near that area.
Also in order to help with the transition, it's best that you keep the lid of the nest box open till momma budgie gets more used to her new and improved nest. Later on, you can very slowly and gradually close the lid of the nest box.
Unless the food cup is very cramped already and there is the imminent danger of the newly hatched chicks to fall off to the cage's floor, then you can move them.
Ideally, it would be best to wait at the very least till the chicks first double in size (age 4 to 5 days old). At the stage the chicks are currently on, they need the constant heat source provided by the mother and the feeding schedule is also very demanding. If this change is too stressful for the mother, she may take more time in accepting the nest and tending to the chicks or worst case scenario, to abandon them.
Before moving the chicks, make sure to wash and dry your hands and warm them up to do the very delicate move to the nest box.
Best of luck with everything and be sure to update us on your growing budgie family.