As the budgie was in quite a state when you found it, I would take it to an avian vet ASAP.
I understand your reasons for waiting, but not knowing how long the budgie has been outside and with the problems you have mentioned, going to an avian vet now would be best.
Please make sure you get the bird to an avian vet, rather than a regular vet. You can call your practice and ask before taking the budgie in, or do an online search of avian vets.
There's a difference between a vet that will see a bird and one that's been trained to treat them.
Why is seeing an AVIAN vet so important?
While you wait for an appointment with the vet, you can try soaking some millet in pedialyte Guardian Angel // Pedialyte // Pedialyte Recipe
Cover three sides of the cage to make her feel safe and keep her warm.
You can also post some details about where you found this bird in the section here.
Lost and Found - Talk Budgies Forums
And these screening questions will also be helpful if anyone makes a claim to ownership.
Found Budgie? - Screening questions
If you don't find the owner, are you planning to keep this bird, or find a new home for it?