Beyond the Apocrypha?

The simplest answer, I suppose, is that they were included in the Bishops Bible (and most previous English Bibles.)

But that begs the question. The books were included in many editions of the Vulgate, and I assume that is how they passed into the English Bible. The Council of Trent omitted them despite their long usage (though there seems to be disagreement about why they were omitted and whether it was intentional), although they were included in an appendix in the Clementine Vulgate (1592). It seems that most English Bibles picked them up because they were in the Vulgates the English had been famliar with.

 

Although the books were omitted from deuterocanon of the Sistine Vulgate, they were included as an appendix (apocrypha) in the Clementine Vulgate (1592), which is the same way they appeared in the original Douay-Rheims, possibly because Clement ordered, according to Scrivener, that every edition of the Vulgate "must be assimilated to this one, no word of the text may be altered, nor even variant readings printed in the margin." That didn't quite work out because Challoner dropped the three books when he revised the DR, which pretty much eliminated them from the Latin Rite's English Bibles. (Not to mention that the text of the DR does not rely on the Clementine edition exclusively.)

My understanding is that the DR actually was translated circa 1580 (before the Sistine and Clementine Vulgates) and I don't know if the three books were translated at that time or if they were translated later to conform to the structure of the Clementine edition.