Then it might work in your favour in this case, because my understanding is that when the father isn't listed on the birth certificate, it is a simpler process to apply for the passport, although looking at the form, it still asks if you have tried to contact the father and obtain written permission to apply for the passport, along with their last known contact details, so I think the very first step is to immediately try to track him down to obtain permission and ideally his signature on the form. If you want to travel in 4 months, get on it right away. The passport office website says that you should allow 6-8 weeks for a child's passport and it might end up taking longer for complicated applications like yours.
Ultimately, have a proper read of the page I originally linked to because really I'm just passing on the same information I'm reading from there myself.
