I can recommend some earlier answers on here for you, but in short there was never one point when the original letters were discovered, they were just continuously copied throughout Antiquity and the Middle Ages: In this thread u/ecphrastic (whose 'cake-day' is today, congrats!) writes about the letters generally but focuses on the one about Pompeii. When it comes to how the manuscript tradition you can also read from u/KiwiHellenist about the works of Pliny the Elder, and from u/LegalAction about the letters of Cicero

Edit: I forgot to say, ecphrastic does mention that the letters to Trajan were "discovered" in the 15th century, but this means that they became widely read and eventually published having only a few copies before, not that someone found originals in Pliny's hand from Antiquity