thank you for sharing your question with the Wefugee Community!
From the little information I have about your case I think it would be best if you definitely ask again at the Ausländerbehörde and at least make them tell you how long it is gonna take them. If they don´t respond or keep on saying that they still are working on it, it could be helpful for you to go see a lawyer or if you don´t have any money a counselling centre for free. Via this website you can find one close to you: https://adressen.asyl.net/language/en/welcome-to-informationnetwork-asylum-and-migration/
Maybe they can ask at the Ausländerbehörde again and find out more.
I wish you all the best and hope that the situation clears up soon and your husband can start his Ausbildung!
Note: Please keep in mind that we can´t provide qualified legal advice via the internet and that any information given can´t replace such legal advice.