Hi, Mith. I agree with you if the Latin equivalent of "with no delay" or "immediately," which is the word used in the English translation, appeared in the Latin text. However, I checked my copy of BD in my later edition of Denzinger which has the Latin text (picture attached), and the operative word in Latin seems to be "mox."
I'm no Latinist by any means - I can only pray the Rosary in it and have some knowledge from the Missal and my interlinear DR/Vulgate New Testament - but when I checked available Latin dictionaries it seems that "mox" can mean merely "soon" or "shortly thereafter," and not necessarily "with no delay" or "immediately."
Perhaps Lad or someone who knows Latin can comment.
Yes, "mox" means "soon", but I think the context makes it clear that he means, basically right away, perhaps after their particular judgment, etc. In the passage about those who are saved, he distinguishes between those who have no sin left and those who requires some purgation, using the term "mox" of the former, so I think that immediately or right away would be a good translation. It has the sense of "the next step for them is ...".