Hi @Sidra,
There is unfortunately no specific time frame for the processing for an application for an Ausbildungsduldung. I have found one document about it from the Refugee Council in the region of Saxony-Anhalt where it said processing was on average 3 months, sometimes longer. If you have submitted a passport and your Ausbildung contract, I don’t see any reason (other than perhaps a large caseload at the Ausländerbehörde due to the situation in Ukraine) for it not to be processed quickly.
Under German administrative law, if an application is not processed within three months, you have the opportunity to submit a legal complaint called an ‘Untätigkeitsklage’. This would be something to consult with your lawyer about doing. Here is some information about this type of procedure. It comes from Wikipedia via an online translation so it is only to give you a basic outline; it is not a reliable legal source:
https://de-m-wikipedia-org.translate.goog/wiki/Unt%C3%A4tigkeitsklage?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp
Best,
Éanna