The Catechism continues (emphasis mine):
404 How did the sin of Adam become the sin of all his descendants? The whole human race is in Adam "as one body of one man". By this "unity of the human race" all men are implicated in Adam's sin, as all are implicated in Christ's justice. Still, the transmission of original sin is a mystery that we cannot fully understand. But we do know by Revelation that Adam had received original holiness and justice not for himself alone, but for all human nature. By yielding to the tempter, Adam and Eve committed a personal sin, but this sin affected the human nature that they would then transmit in a fallen state. It is a sin which will be transmitted by propagation to all mankind, that is, by the transmission of a human nature deprived of original holiness and justice. And that is why original sin is called "sin" only in an analogical sense: it is a sin "contracted" and not "committed" - a state and not an act.
The Church does not profess to know when and how the newly-created soul acquires original sin; however, it is inextricably bound to human nature, so one would expect it to be acquired as the soul acquires its human nature, although in a very real sense the soul of a person is itself their human nature.
All we know is, Adam sinned, and his sin cast a shadow upon our nature and gave us a certain tendency to despise God and love evil. This part and parcel of being human is present in us when we are born, and we have no Revelation to guide us any further.