Welcome to our community and thank you for your question!
First of all, unfortunately you will most likely not get a residence permit through your asylum application in Germany. In accordance with the so-called Dublin III regulation you can only apply for asylum in one EU country. In your case that’s Italy. German authorities will therefore refer you back to Italy.
You can try to get a residence permit through your partner though. She is a EU citizen. That’s why marrying her grants you a residence permit and the right to live/work everywhere in EU.
I’m not familiar with British authorities and procedures. Maybe your partner can try to get in contact with counselling centers to find out what options you have, for example concerning acknowledgement of paternity and child custody. In Germany that’s a way to obtain a residence permit for non-married partners. There might be similar (or other) options in the UK.