This is somewhat tricky to nail down. While Galileo is credited with being the first to mathematically define speed, he was not the one to actually coin the term speed. However, back in 375 BC, Eudoxus wrote a book about speeds, and so it is to him that the credit must go. There is a caveat to this - speed was already a defined term - Eudoxus was the first to actually write it down.
The ancients were well versed in speed and the many benefits of it. The Egyptian, Babylonian, and Akkadian nations were far advanced in math and sciences, and we are still discovering new information to show just how advanced they were and what they actually knew. With the burning of the library of Alexandria, important documents were lost to the world forever, much to our discredit.
Speed would have been used to calculate movements for armies, shipment time frames, camel trains, and so much more. Speed was not a new term, nor was it a new concept. It may have been called different names over the millennia.