If the birds have been successful in raising their young in a nest box, the babies will fledge and then there is at least a two week break before they might begin to raise another brood. I always call it their family vacation time. You can clean the nest box at this time while the baby birds are shown the territory and taught how to forage on their own.
If something happens to disrupt the success of the first batch, the birds might begin a new nest within a week. You don’t have to remove the nest in this case but broken eggs or dead nestlings should be removed immediately. If they want to try again in that box, it will give them a head start to have an existing nest. Also try to determine why there was a failure and how to prevent further tragedy.
By cleaning out a nest box you help deter parasite infestation and a predator’s ability disturb a nest that is built on top of old nests making it closer to the entrance hole.
To clean the nest box I usually place a plastic bag over the nest and just sweep it all in and twist the bag shut. You can rinse out the house with a water hose or diluted bleach spray. Make sure the drainage holes are unplugged and leave the house open to dry for a couple days. Finally dispose of the old nest in the trash and wash your hands thoroughly.
Taking care to clean your feeders and nest boxes makes you a responsible steward of nature. Thanks to Barbara S. for a very good question.