LANL published a paper circa 2007 or 2008 on how to make carbon-neutral hdyrocarbon fuel using carbon in the air and the electricity and waste heat from a nuclear plant. According to the paper, it was competitive with oil-based gasoline at around $5/gallon.
Its authors badly underestimated how much new American nuclear reactors would cost.
We performed economic analyses on a partially optimized baseline concept based on a single Gen III PWR to provide power for the process. The analyses estimated a capital cost of $5.0 billion for an 18,400 bbl/day synthetic-gasoline plant and $4.6 billion for a 5,000 tonne/day methanol plant. Nuclear power accounts for more than 50% of the total plant capital investment.
If just over 50% of that cost is supposed to be nuclear plant, it would be $2.5 billion for a Gen III pressurized water reactor. ($2.9 billion now, accounting for a decade of inflation.) The just-abandoned VC Summer project was supposed to build two new Gen III PWRs in South Carolina. They pulled the plug because estimated cost of completion had spiraled up to $25 billion ($12.5 billion per reactor).
You don't need gasoline. The gasoline ICEs can run on ethanol and LPG and diesels can run on DME (dimethyl ether) and vegetable oils. DME can be produced using nuclear process heat and/or hydrogen or using biomass.
it is called battery charging :). Basically it is making the fuel - Li - out of the ash - lithium oxide. The amount of Li and available for "burning" oxygen in the current batteries is very small and where is no principal reasons it can't be increased 10-fold to something like 10% by weight (especially in the metal-air schemes) at which moment it will leave gasoline engines well behind in all respects.