Even with base spell speed and 0% cast time reduction, Enochian -> Thunder III -> Blizzard IV is at most 6s out of the 13s Umbral Ice timer.
Press the button for the next spell before the current spell finishes casting, so that you begin casting the next spell immediately. As a BLM, your DPS relies on pumping out spell after spell without any gaps between spells. This will be a simple matter on a dummy. In real situations, you'll need to figure out how best to fit your spells into the current fight's mechanics and boss rotations so that you can continue to dish out all the splodies with minimal down time. Keep practicing on the dummy until you can press all the buttons without having to think about them, so that when you're against a live target, you can focus on the mechanics instead of the rotation.