Hi @sunnycity01
There is no specific timeframe for an objection letter (Widerspruch) to be answered. Only after three months without a response can you make a complaint of inaction on the part of the Ausländerbehörde. This is called ‘Untätigkeitsklage’.
In urgent situations, it is possible to file a legal suit against an authority before the decision on the objection letter is made. This is called in short ‘Eilantrag’ and in full ‘Antrag auf Gewährung einstweiligen Rechstsschutzes‘. You can file this with the Administrative Court in Stuttgart. In the application, you have to describe why it is urgent, i.e. if the lack of a residence permit is affecting your ability to take part in the Ausbildung. I recommend you obtain legal advice before submitting such a lawsuit.
Please note also that as you did not enter Germany with the correct visa, you can only be issued with a permit in exceptional cases. Hopefully in your objection letter you demonstrated that it would be excessively unreasonable to require you to leave Germany to obtain a visa. The fact that you have an apartment registered here is not relevant. The fact that you have begun the Ausbildung may work in your favour but it really is only in exceptional cases that they ignore the visa requirement.
Best,
Éanna