No biggie.

Hopefully the mother is insured with a proper assistance company,for them that's actually daily business. I had multiple calls when we were tasked with flying home a mom and preterm babies or two. One with neither mom and dad knowing beforehand a pregnancy existed.(that was kind of funny, though,as both had a medical background and were pretty cool about it)

From the document side of things for the industrial nations and the more developed destinations it is usually also no problem - first a local certificate of the birth is required,the closest embassy/consulate is contacted and emergency travel documents are created. As the old children's passport have been phased out by virtually all countries are regular provisionary passport is granted (that is not printed abroad but filled out by the embassy/consulate) and you are ready to go.

Only intermittent stops/layovers can be a problem sometimes, especially in some countries (US....) so with a professional assistance company one takes care to avoid these. Furthermore some countries do not trust birth certificates from some countries (e.g. due to a high number of surrogate motherhoods) and may demand further proof of the motherhood - e.g. medical records from antenatal care back home or a medical certificate of a recent birth by one of the embassies trusted doctors. But these problems can usually be solved easily.