Welcome to the forums!
I'm sorry for your current predicament, mother budgie is at a great disadvantage without the help and support from the male when it comes to feeding her.
In this case she has to do all the feeding. Place millet sprays near the entrance of the nest, the same goes for food and water containers and egg food, she must have easy access to all this so that the chicks don't get too cold when she leaves the nest to feed herself.
How many eggs do you currently have? Can you show us a picture of how the bamboo nest looks like? If the chicks are falling out, this type of nest is not at all safe for them to grow in. If the chicks are less than a week old, let them stay for now on that nest and place a soft blanket directly below the nest in order to cushion that area in case a chick falls down.
During this time you need to pay close attention to the chicks development, to see if they are growing well and not developing splayed legs. Chicks can develop splayed legs when they are in a less than desirable nesting area that does not give them proper traction and as they grow, their little legs spread apart and if not corrected in time they end up disabled. This is why it's so important to have a nest box with a good concave bottom.
To avoid this from happening, you really need a proper nest box for breeding budgies and make sure it has a good concave bottom. Also get a brooder, hand feeding formula and syringes in case the move to the proper nest box doesn't go well and momma budgie abandons the chicks. Even if all goes well, you must need to be prepared to give an extra help to the chicks, because they don't have their father when they reach weaning age and if the female turns against them, you will need to separate them from mother and help them to wean and gain independence.
Be sure to read all the info on the links Deborah has provided in her post, so that you are better prepared to give all the necessary help to mother budgie and her chicks.
I would also like to inform that it's not at all advisable to allow a rescued budgie to breed. Without having an idea of the correct age of a rescued pet bird nor knowing its past history, allowing said bird to breed when only a couple of months have passed by after the rescue is not the best way to proceed and very sadly you had to learn this the hard way.
A budgie pair must be in top physical health and condition and be under a diet appropriate for breeding before going through the task of mating, laying eggs and raising their young. They must be older than 1 year old and not older than 4 years (5 years for the male under special circumstances).