thank you for sharing with the community the situation you are facing.
After having done the Vaterschaftsanerkennung, you have to get the "Erklärung für Sorgerecht" you do that at the Jugendamt- this certifies that the two o you are taking care of the child- shared custody.
After that you send in everything to the Verwaltungsgericht, BAMF and Ausländerbehörde.
It might take a while but do not worry, in order to speed up the best would be to ask every 2 weeks how long it might take.
I hope we could assist you with our answer, if you need further help please do let us know.