Hi @Adusepoku88,
Firstly, we are not lawyers on this site and so cannot provide legal advice. I can only offer some information about the law based on the information you have provided.
A question from me is: on what basis is your friend being promised a place to live after she leaves hospital? On what basis are the costs for this being covered? Who has covered her medical costs, presuming she does not have German health insurance?
I ask this because generally the social welfare authorities have to clarify a person’s immigration status before granting them assistance in the form of housing or money. They will only assist those who are entitled to such help.
I am assuming the father has a long-term residency in Germany after Brexit. This is called Aufenthaltsdokument-GB and should be written on his residence permit (Ausweis).
You mentioned the Vaterschaft. Did the parents also do a declaration of shared custody (in German ‘Sorgerecht’)? If so, there is a better chance of the children receiving a residency status due to his status. He would have to show he is able to provide for them financially and has regular access to them.
For the mother it is more difficult. For her to then get a permit based on the children as their family member she would have to show that otherwise it would be a case of extraordinary hardship. Alternatively, she could request a permit based on humanitarian reasons. There may be a situation where she is issued with a Duldung (not a full residence permit) for humanitarian reasons on account of the children.
Due to the way she came to Germany, it is likely she will be told to return to her home country and apply for a visa as a family member at the German embassy there. However, this is not correct. As the parent to three newborn children, she cannot be expected to leave them in order to go through a visa process. The local Ausländerbehörde have to process her residency application here.
She can get more detailed information and assistance in dealing with authorities from a migrant advisory service. Here you can search for one in her area:
https://bamf-navi.bamf.de/en/Themen/Migrationsberatung/
Best,
Éanna