Working on a future history setting; the question arises, since the world cannot run on fossil fuels forever, one way or another, something must end up replacing them; what will that something be?

A co-author has suggested using nuclear energy to electrolyze water for hydrogen which is combined with carbon dioxide to produce methanol. The chemistry of this is sound but I'm not sure about the economics of getting hydrogen that way. As I understand it, hydrogen produced by electrolysis currently costs several times as much as that produced from natural gas, which suggests methanol produced by this method will always be at least a few times more expensive than current fossil fuels (assuming future nuclear energy will never be as cheap as current solar for applications like this that don't need external storage); it seems likely to me that the world will be very reluctant to accept a several-fold increase in the cost of fuel. (I'm not talking about what should happen, but what plausibly will happen.)

On the other hand, it turns out to my surprise that there already exists a facility making methanol by the above method except geothermal substituted for nuclear: https://www.chemicals-technology.com/projects/george-olah-renewable-methanol-plant-iceland/

On the third hand, that article does not discuss the economics of the hydrogen source; for all I know, maybe it's a pilot project built as proof of concept that is not, nor expected to be, economically competitive.

I can't find a clear answer as to how much hydrogen costs from various sources. Google finds estimates all over the map, ranging from a dollar per gram to a dollar per kilogram.

So my question is:

Assuming advanced nuclear reactors can provide a reasonably cheap (and clean and safe enough to satisfy political requirements) source of heat, and the technology is mature and acquires relevant economies of scale, how much would electrolytic hydrogen end up costing? Either in dollars per kilogram, dollars per joule, or relative to the price of oil or natural gas?

Spoilers

These estimates handwave a lot of technology that does not yet exist, but people are actively working to build. I think I’m playing fair by the rules of hard science fiction. Don’t base any investment decisions on them!

If we could build the nuclear reactors as cheaply as South Korea manages to in the real world, and gave them no other subsidies, we could bring the cost of the hydrogen in a gallon of methanol down between 72 cents and a dollar. The total cost of all the raw materials, using captured carbon dioxide rather than fossil fuels, might be $1.14–1.41. Since the process is not perfectly efficient, we might actually end up spending $1.34–1.91.

Since one gallon of gasoline has twice the energy content of one gallon of methanol, we should double this: replacing a gallon of gasoline with methanol would consume gases that cost $2.67–$3.83 to produce, unless a more-efficient process is found. But, the question specifies hydrogen production as a separate step.

An amount of hydrogen equivalent to a gallon of gasoline, if you used it directly in a fuel cell rather than making methanol from it, would cost $1.88–2.66.

For comparison, the price of an energy-equivalent amount of methanol has fluctuated in recent years from $1.08 to $3.00.

The bottom line here is that this process would not be commercially-viable without a subsidy, unless the price of natural gas rises. It’s not a several-fold increase in the cost of fuel, though, but a modest one. The most plausible way to change that would be to discover a better catalyst to produce either carbon monoxide or methanol from water and carbon dioxide.

Building the Plants

The best-case scenario for nuclear power construction is South Korea, the only country in the world where the cost of building nuclear plants is falling over time. (However, even there, its current government plans to phase nuclear power out.) Recent plants there cost 2,000,000 KRW/kW. This is equivalent to an “overnight” construction cost of $2,000/kW. The World Nuclear Association, admittedly not a neutral source, estimates that, at a 3% discount rate, this equates to a levelized cost of electricity of $29/MWh. If the discount rate increases to 7%—that is, the investors need the plant to be paid off faster—that increases to $40/MWh. (Note that the high range of the estimate in @kingledion’s excellent answer was based on an electricity price of $80/MWh and a cost to nuclear energy of double that, while the low end of both estimates is approximately the same.)

Could the South Korean cost be replicated? To some degree, maybe. A major reason for the savings is that they’ve built the same design over and over, and this scheme calls for a lot of standardized reactors. The reactors could also be built wherever the economics are best, and export the fuel to other countries.

Generating Hydrogen

However, the point of using nuclear energy for this is that it can beat the alkaline water electrolysis technology that the US Department of Energy assumed, and reduce the cost by even more than the 50–62.5% that simply using the real-world cost of nuclear power in South Korea would suggest.

What this plan actually calls for is for a reactor like the Very High Temperature Reactor or the Gas-cooled Fast Reactor, which both work on the same principle, except that the former uses the uranium fuel cycle and the latter a fast-breeder reactor. Both are sealed, integral reactors that use supercritical helium as both the coolant and the working fluid to drive the turbines. For this application, you would want to run the reactor at the supercritical temperature of helium, 850°C, and the high-temperature steam hydrolysis reaction around 800°C. The cost of construction could be further reduced if we ever figure out how to use carbon dioxide in place of helium.

The current state of the art in hydrogen production can achieve energy efficiency of up to 50% (by high-temperature steam electrolysis, or other methods such as the sulfur-iodine process). In other words, since the energy density of hydrogen is 33,300 Wh/kg, a thermal reactor using its process heat plus a voltage of about 1.2V could potentially achieve a yield of $0.5 × left( frac{3.330·10^{-2} mathrm{MWh}}{mathrm{kg}} right)^{-1} = 15.02 frac{mathrm{kg}}{mathrm{MWh}} $.

If we naïvely (excuse me, optimistically) handwave that you can get power from a GFR as cheap as South Korea gets it from third-generation reactors in the real world, that is, $29–40 per MWh, that cost would fall to $1.93–2.66 per kilogram of hydrogen. Most of this can be thermal energy used as process heat with no conversion or transmission losses. It’s probably not realistic to pull that off, but it definitely is possible to do better than $80, the basis for the high-end estimate in the DoE report.

The DoE report assumed a cost of $3.00/kg.

Synthesizing Methanol from Hydrogen and Carbon Dioxide

This technology does not require a high-temperature reactor and more likely runs off clean electricity from the grid. It is in use in the real world already, but with carbon dioxide captured from sources such as coal plants.

Each (US liquid) gallon of methanol contains
$$ begin{align*}
frac{3.785·10^3 mathrm{cm}^3}{mathrm{gallon}} &× frac{0.7920 text{g CH₃OH}}{mathrm{cm}^3} \
&× frac{1 text{ mol CH₃OH}}{32.04 text{g CH₃OH}} \
= frac{93.56 text{ mol CH₃OH}}{mathrm{gallon}}
end{align*}$$

Hydrogenating carbon dioxide to produce methanol needs two moles of H₂ and one mole of CO₂ per mole of CH₃OH. (If you convert the carbon dioxide into carbon monoxide.)

In practice, the reaction would be more complicated: either it would consume 50% more hydrogen (the reaction 3H₂ + CO₂ → CH₃OH + H₂O turns the extra hydrogen back into water, which can at least be recycled back to hydrogen) or it would replace some of the hydrogen generation with a process generating hydrogen and carbon monoxide from water and carbon dioxide, whichever were more efficient.

Since the molar mass of H₂ is 2.014 g/mol, if we use 3 moles of hydrogen gas per gallon of methanol, each gallon of methanol requires 0.5654 kg of hydrogen as input. Using the numbers in the previous section, the hydrogen used to produce a gallon of methanol would cost between $1.09 and $1.50.

The molar mass of CO₂ is 44.0095 g/mol, times 93.56 moles, so we need 4.118 kg of CO₂ to produce one gallon of methanol. The idea is to use captured carbon dioxide, rather than fossil fuels, for example from incinerating biomass or direct air capture. The existing Climeworks AG plant in Zurich captures carbon dioxide from the air at a cost of $600/tonne. If hypothetical technology could bring the cost down to $60/tonne, which many sources believe is possible, the carbon and oxygen in a gallon of methanol would cost 24.7 cents. A more conservative estimate of $100/tonne would cost 41.2 cents.

This gives us, using assumptions about cost that are optimistic but somewhat plausible, a cost of $1.34–1.91 for the raw materials. Since one gallon of gasoline contains as much energy as two gallons of methanol, double this to get an estimate of $2.67–$3.83 per equivalent of a gallon of gas. This is just to produce the raw materials; there would also be a cost to synthesize the methanol, blend it, and distribute it.

Subsidies

However, this assumes that the governments of the world do nothing to subsidize green fuels or to tax carbon-dioxide emissions. If they start taking climate change—and energy independence—so seriously that they would be willing to totally reverse course on nuclear power, that seems unlikely. Someone has to make energy producers pay the external cost of the pollution they create, or else it will always be cheaper to use the same nuclear reactor to make methanol from fossil fuels.

If a government were willing to simply eat the cost of a hydrogen plant that could produce megatons of clean domestic energy, the way they’re willing to eat the cost of a war in the Middle East, or subsidize it like they subsidize oil and biofuels, the marginal cost to produce hydrogen would fall drastically. If the hydrogen did not have to pay back the capital cost of the reactor, just break even on operating cost, the marginal cost to produce hydrogen would be tiny.

Although you said in a comment that the methanol is meant for use as a transportation fuel, if it were turned into something that isn’t burned, such as olefins or formaldehyde, it would remove and sequester large amounts of carbon from the atmosphere.

The reactors would also produce other products that generate profit, such as electricity. Or, for example, since electrolysis of water produces half as many molecules of hot oxygen as hydrogen, it might react that with phosphorous to produce food-grade phosphoric acid.

Alternatives

If you already have hydrogen and captured carbon dioxide (from an incinerator or any other source), you can make methanol at relatively low temperatures. One alternative would be to increase the amount of hydrogen by 50% and use a catalyst such as Ni-Ga at 200°C. Another is co-electrolysis of hydrogen and carbon monoxide from carbon dioxide and water. Another is splitting carbon dioxide and water through a thermochemical cycle.

It would also be possible to convert wood or other cellulosic biomass to syngas through destructive distillation, and then turn that syngas into methanol through conventional means. In fact, methanol used to be called “wood alcohol” because that was the first industrial process to synthesize it. I’m inclined to assume that’s not very economical, since the technology was abandoned decades ago, and I’m not aware of any serious feasibility study. This report by the Methanol Institute (again, a biased source) at least mentions making methanol from wood as a possibility.

The cost of the inputs might also be brought down if the captured carbon dioxide came from emissions that otherwise would have entered the atmosphere, as in the municipal trash incinerator in Oslo that captures its carbon dioxide for use by the oil industry.

$3-6 per gallon of gasoline equivalent (gge)

In 2009 the National Renewable Energy Laboratory assembled a panel to generate 80% confidence intervals for the future costs of hydrogen production. The report is here. This panel used technology available in 2005, but capitalized with a full industrial infrastructure, to make the best use of production scale (this is the 'central production model' discussed in the report). They panel focused on two technologies: alkaline water electrolysis and polymer electrolyte membrane electrolysis.

A kg of hydrogen is referred to as a gallon of gasoline equivalent, since it has roughly the same amount of energy as a gallon of gasoline.

The average cost was strongly dependent on the cost of electricity. The assumed 'high' price for electricity was $80 per MWh, whereupon hydrogen costs are $4.78 per gge. However, the levelized cost of nuclear power is currently around $120 per MWh. If we introduce this increase in electricity power, we get an assumption of more like $6 per gge in costs.

The low costs for electricity at $0.03 per kWh, and all other factors minimized would be about $1.70 per gge.