How exactly does the impregnation work do i have to wait a few in game days or is there something else

Once it happens, you can spam the sleep button to get results.

Its only visual and it works over time, the sleep button just speeds it up.

What happens to the children after they are born? Are they sent to a nursery?

No, it is just a visual difference Faun did not add a plot to the pregnancy.

Yeah, I think he needs a plot that parallels the harems of the Ottoman Empire where the main purpose was to generate heirs to the empire. The current idea is to convert the island population to worship the goddess.   Here's a series made in Turkey on the sultan's harem "Magnificient Century". Women seem to love the series.