Hello and welcome to TB, What your hen did is completely normal this is how they will encourage the chicks to become independent and she could quite possibly be coming into condition once more. Do you still have the nest in the cage? I would recommend you take it out as she needs to recuperate from rearing her first clutch of babies.
Make sure she has lots of good quality food available , greens, egg and biscuit mix is good as well and of course calcium bells a must.
As far as the young budgie becoming sick unless you had a necropsy done then we would be purely guessing I am afraid.
Was he puffed up and sleepy , making any noises, sneezing a lot, what was his droppings like? As you can see the reasoning is very hard to fathom.