Eucalyptus oil is a strong scent and I think the feather mites do not like it. The oil also has natural antiseptic properties - but a drop is all that is needed as it is strong and concentrated!
It won't harm the budgies - they nest in Eucalyptus trees naturally and the scent will also help them with keeping their airways clear.
A tiny bit of olive oil will help smother Scaly face mites
, which are totally different to feather mites
If you think the babies have feather mites, pop them on a white piece of paper and see if any little moving specks appear. If they do, you could gently stroke them off and have bits of small pieces of sticky tape ready, to trap the rotten mites and squash them when you're finished.
I used this method on a 5 week old parakeet (A Rubino Bourkes) because she was too young to use Ivermectin. I got the lot and she did not need medicating!
And now I need a shower because I am so itchy, it is ridiculous~