Hello Salome Mkota
I'm very sorry for answering late, I have really much work now and I've overseen the request.
First of all, at least 8 weeks after the birth nobody can deport you anywhere, even if you and the baby are absolutely healthy. It is so called mother protection.
To the question of Austria: In case of asylum seekers the responsibility of the state, which issued a visa, to take the person back is finished 6 months after expiring the visa. After that the state, where the person actually stays, will be responsible for proceeding the asylum application.
As far as I understand, you made no asylum application but just stayed illegally. Such cases will be regulated not on the basis of Dublin Rule but on the basis of agreements between two countries.
I suppose that perhaps Austria isn't responsible anymore. Then Germany can only try to force you to go back to your country of origin. But you have naturally a strong argument against it, if your child has German citizenship. (Certainly having a child whose father lives in Germany also prevents a deportation to Austria.) As a mother of a German child you have a right to stay here as long as you excersise your child custody.
As soon as you have documents about the German citizenship of the child, you can apply with the Ausländerbehörde for a residence permit after §184.108.40.206 Aliens Law (Aufenthaltsgesetz). If it takes time, the Ausländerbehörde must first at least give you Duldung (Toleration) but they also can give you a Fiktionsbescheinigung (a document that you applied for a residence permit and your application is under consideration).
Sometimes the Ausländerbehörde treat people like you in a bad way and insist on going back to your country and make up for a visa (as family reunion to your German child). The basis of this claim is a special paragraph in the law: normally a foreigner must come to Germany with a proper visum to get a residence permit. But in your case (giving birth to a German child) an exclusion is possible and the well-being of a German child who needs a mother and can't be separated from her (in this age even for a short time) is a very strong argument, with more weight than a missing visa procedure.
It is good that you have a lawyer. It normally makes things easier for you.