My guess is they'd planned to leave the UK for Republic of Ireland via Liverpool before the baby was born, but either they'd got their dates wrong (easy to do with no antenatal care) or the baby came a bit early. Going to Ireland to escape social services is a fairly well trodden path as the perception is they're less likely to remove children.
www.irishpost.com/news/british-families-flee-to-ireland-to-escape-threat-of-social-services-63574
However, it all went wrong with an earlier than expected labour and the car catching fire (known to be a mechanical fault) and sending their belongings up in flames, possibly including her passport (he's known not to have one). Travelling to Ireland by ferry doesn't need a passport especially if you're going via Belfast.
I hope someone is harbouring them because if not the baby is sleeping rough. If they are sleeping rough, and they've got any sense, they've probably set up their tent in a disused barn or other outbuilding so they're not noticeable to passersby.