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?

$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.