I would bet on price going down slightly with scale, but one can't really tell now what will happen: it might go up a lot, it might go down a lot, or it might stay flat.
Batteries have the advantage of being explorable at a small scale. Now that the potential market has become so clear this is happening, in many companies.
And uranium seawater extraction already exists: https://www.forbes.com/sites/jamesconca/2016/07/01/uranium-s...
It's more expensive than mined uranium, but since fissile material is so energy-dense that increase in fuel cost amounts to hardly any change in overall cost.
Seawater uranium extraction is at a much lower TRL (technology readiness level).
This is an excellent example of your hypocritical double standards on this subject.
You insist that hydrogen is so technically ready, yet nobody is using it.
Dude. You are falling back to the "if it isn't already being done, it can't be done" argument. Please stop this foolishness.
Hydrogen is being stored in a few places. That the storage isn't larger isn't because of any technical obstacles, it's because there's no reason to store it now. In particular, when we can burn natural gas without CO2 charges, using the hydrogen for energy storage is pointless.
This doesn't mean hydrogen CAN'T be stored, it just means the market conditions for widespread adoption of an off-the-self technology aren't there yet.
It's not just a question of storage, you can just use a salt cavern for that.
It's also a question of electrolyzing water into hydrogen efficiently.
And converting it back into electricity efficently.
And building all of these systems cheaply.
And deploying all of these systems at massive scale.
We're still on the first phase of that. As per your other comment we still don't even have effective elctrolysers to do this cost-effectively [1].
Will hydrogen storage pan out? Maybe. But until then it's not a solution. It's a potential solution, like fusion, or algae in vats, and thermal storage, and all the other potential solutions being proposed. It's not a solution that has actually demonstrated viability.
Why shouldn't nuclear plants scale? They're mostly just steel and concrete. Uranium is more than 40 times more prevalent than gold, and it's energy density is such that it represents a negligible cost of operations. The technology is just scaling up existing components, we had nuclear powered submarines for a while. This is what people thought about nuclear power in the 1950s and early 60s. As plants actually started being constructed problems such as corrosion, large amounts of earth moving, metal impurities, and more were discovered and made the plants more expensive.
We haven't discovered these issues with hydrogen storage. We won't discover these issues until we actually build hydrogen storage facilities at scale. We don't know what challenges will lie in store when building hydrogen storage, because we've never done it before. This is why it's useless to talk about the cost of hydrogen storage until we actually have experience building and operating hydrogen storage plants. Our knowledge of cost of hydrogen storage is in the same situation as nuclear power in the 1950s.