Most tablets were air dried, which would take a day or two. But some tablets were fired - dried in a kiln. I don't know about Hittites but at least in Mesopotamia, air drying was by far the most common method and only some important ones were fired. So methods were certainly available to dry a tablet quickly if necessary but I doubt if air drying delays made much of a difference. Messages and goods were limited to the speed of horse after all.