In classical Latin word-final vowels before initial vowels were elided also in prose, not just in poetry. However, I am under the impression that elision is not (intended to be) used in contemporary Latin prose. I am not aware of a Latin pronunciation practice currently in use where elision is applied. Prose of all eras is read without elision in today's Finland, and I doubt we are special in that respect. If this observation is correct — please correct me if I'm wrong — then elision must have fallen out of use at some point in the last two millennia. Do we know when this happened? Any more details on the process would also be interesting, but to get started, I want to stick with a time estimate.
The time is unlikely to be sharp. Elision is not binary, as one can easily elide in some of the cases. The practice seems to have shifted from "elide (almost) always when possible" to "elide (almost) never even if possible", and this shift may have taken some amount of time. The shift may also have been regional, as there are and have been many schools of Latin pronunciation across Europe and I don't know how uniform they were in this respect. I still imagine that it is possible to date the phenomenon, or at least some key symptoms thereof.