‘Who made the flood?’ asked Frodo.
‘Elrond commanded it,’ answered Gandalf. ‘The river of this valley is under his power, and it will rise in anger when he has great need to bar the Ford. As soon as the captain of the Ringwraiths rode into the water the flood was released. If I may say so, I added a few touches of my own: you may not have noticed, but some of the waves took the form of great white horses with shining white riders; and there were many rolling and grinding boulders. For a moment I was afraid that we had let loose too fierce a wrath, and the flood would get out of hand and wash you all away. There is great vigour in the waters that come down from the snows of the Misty Mountains.’
- Elrond commanded the flood.
- The river is under Elrond's power.
- It will flood when Elrond has great need to bar the Ford.
- Gandalf added white horses with shining white riders and rolling and grinding boulders.
- Gandalf was afraid that "we" had let loose too great a wrath.
- The waters from the mountains contain "great vigour."
This seems pretty clear that Elrond is in charge here, and isn't just invoking Ulmo. If Gandalf saying the river is under Elrond's power means Elrond's power to call upon Ulmo, the text doesn't imply that anywhere. The river does have vigor of its own, but Elrond is not passively counting on that. He commands the river.
How this works is important to answer the original question: how to represent, with game mechanics, servants of the Shadow trying to cross water. If the difficulty is in whose power is in the water, then this turns into a roll involving the Attribute of whoever puts forth that power. If the difficulty is inherent to water itself, then it's a static difficulty, possibly set per body of water. Furthermore, not all servants of the Shadow are affected by water in the same way. Orcs don't want to cross Anduin, but that seems to have more of a strategic reason than a water-fearing one.
If Nazgul were driven by players, I'd expect a Fear test against the power-wielder's Attribute + something. But players don't play Nazgul, and neither do they play people who put power into water. That turns it into an NPC vs. NPC kind of thing, which you usually don't need to roll for—just decide what happens. If, somehow, the event involves the actions of player-heroes, then just let them roll whatever appropriate task they call for.