Greetings @OluJacob
We cannot provide legal advice here and so I cannot say what your chances of being successful with an appeal are, besides not knowing the details of your situation. I can say, however, that some of the information you provide is not relevant to an appeal, i.e. that you children are going to school or that your wife is doing an Ausbildung.
What is relevant is solely the grounds you gave for asylum and whether the court would view these as sufficient to provide you with some sort of refugee protection. The one aspect you mention that may play a role is that your child is undergoing medical treatment in Germany. If such medical treatment is not available in your home country, then this may be grounds for receiving the protection status ‚deportation ban‘ (Abschiebeverbot). If this condition was already present, then BAMF in its initial decision and the court must have already previously assessed this.
If you do not appeal or your appeal is rejected, then your wife can receive a special type of Duldung due to her Ausbildung called an Ausbildungsduldung. This does not automatically give you and the children the right to stay in Germany but I would expect the immigration office to issue you and the children with a regular Duldung while she is still doing it. If she successfully completes it, then she can receive a regular residence permit, alongside you and the children as family members.
I am not an expert in the area of deportation but the basic procedure is as follows: If your case is finally rejected, then you are given a deadline to leave the country voluntarily. The deadline is between 7 and 30 days. Otherwise a deportation can take place. This notification is called an ‚Abschiebungsandrohung‘. You are not notified of an actual planned deportation.
In case of a final rejection, then your wife must make an application for an Ausbildungsduldung, and for you and the children a Duldung. In the application for you and your children, you should give her Ausbildung and your child’s medical treatment as grounds for a Duldung.
As you have been involved in court proceedings, then I assume you have a lawyer. Our advice here cannot replace legal advice form a lawyer who knows your case in detail.
Best,
Éanna