Dear @Jj061987,
Congratulations on the birth of your daughter!
The status of your boyfriend does not affect the birth certificate. What can cause the delay is having both your documents checked to verify your identity. This can take some time.
Until the birth certificate is issued, you can ask the Standesamt to immediately issue you with a copy of the birth registration. In German this is called ‘Auszug aus dem Geburtenregister’ and must be issued in a situation such as yours. This birth registration allows you to receive Kindergeld and Elterngeld.
The Standesamt will have informed the Ausländerbehörde about your boyfriend. They will contact him and most likely order him to leave Germany. When this happens, he should send the Ausländerbehörde a copy of the birth registration or birth certificate abnd request a residence permit as a family member. However, he will only be able to stay if he can show his living costs are covered.
The Ausländerbehörde may request him to leave Germany and apply for a visa as a family member to return here. This is not correct as he has a young child and so can be issued with a permit directly here without leaving Germany.
Your daughter is entitled to a residence permit according to paragraph 33 of the Residence Law.
Best,
Éanna