For this to happen, we need a "Mr Fusion" scenario. Conventional fission reactors are pretty massive even without the shielding because of the large and complex machinery to convert the heat of fission into electrical energy. Coal plants are large and expensive for the same reason.
If you could convert the thermal energy directly to work, you can get some pretty amazing efficiencies of scale. The Phoebus NTR simply used thermal energy from fission to heat hydrogen reaction mass, the ultimate version ran at 4000MW for 12 minutes in 1968, one of the most powerful nuclear reactors ever built (the core was apparently about the size of a mini van).
If you are interested in electrical energy, you might consider a "dusty plasma fragment" engine, which injects a critical mass of finely powdered fissile "dust" into the core, where the heat converts it into plasma. As a spacecraft engine, the plasma is allowed to exhaust out the back end with a velocity of @ 1% c. (Actual particle speed inside the reactor is calculated to be .3 c, but internal collisions reduce the velocity considerably). On Earth, the plasma can be (possibly) contained by electrostatic and electromagnetic fields, and the high speed exhaust captured by a MHD generator with high efficiency (theoretically up to 60%, far better than the Carnot limit of @ 40%). Of course you still need shielding, and some way to contain the exhaust.
Better yet are various proposed aneutronic fusion schemes, which use heavier elements than Deuterium (D2) or Tritium. Schemes such as p+Be or fusion involving 3He release much of the energy as high energy alpha particles (Helium nuclei once they slow down). A beam of high energy alpha particles is really a beam of high energy current, and electrical energy can be extracted from the moving beam with high efficiencies (theoretically up to 80%), making for a very compact machine. In the real world, we have not made even "ordinary" fusion reactions take place releasing more energy than goes into the reaction, and aneutronic reactions are much harder to start.
Compact devices like the ones described are still fairly large, about the size of mini vans or shipping containers without various sub systems attached, so you won't see them in cars. Since they produce ridiculous amounts of energy, putting them on "small" vehicles is going to waste a lot of the potential of these devices. You would likely see something along these lines aboard an aircraft carrier, luxury cruise liner or a monster container cargo ship. Scaling for aircraft would be interesting, you have so much energy that you could theoretically power something far larger than an A-380 or C-17 Globemaster III, but now you have the issue of getting a large enough runway and terminal. Maybe flying boats would make a comeback.
The idea of a Grid might become obsolete, with compact reactors sprouting up on the edges of suburbs and industrial parks. With the massive amounts of energy available, these places could be largely self contained, with some of the energy being channeled to recycling materials and purifying wastewater for local reuse. Since each place is self sufficient in energy, power blackouts will be very localized (until you did a reactor restart). At this scale, electric cars will finally become practical, since the electrical infrastructure will be scaled to charge millions of cars by default.
Where it gets scary is when we start using these devices in space. Since anything in space is moving at extreme velocities, any spacecraft becomes de facto a weapon. (to give you an idea, the New Horizon spaceship that visited Pluto is moving so fast that if it passed overhead when you fired a gun from the 0 yard line of a football field, the bullet would reach the 10 yard line as the spaceship sailed over the opposite goalposts). Having the amount of energy that compact nuclear fission or fusion reactors can bring means you could accelerate a ship the size of the ISS for a Mars mission and take as little as 39 days to get there. Your deltaV would be about 9Km/sec (depending on various parameters), but you would have enough energy to easily make that velocity change. It also means you would have that much velocity if you happened to hit anything either by accident or on purpose. Since the magic equation for kinetic energy is ke=1/2MV^2, you can see that increasing velocity allows you to have access to a lot of kinetic energy. People will tend to be nervous about fast moving spaceships for very good reason.
So there will be lots of good and bad effects of having compact nuclear reactors and abundant energy. Second and third order effects will require a lot of thought (cars have been conceptualized since at least the time of Leonardo da Vinci, but no one seems to have predicted strip malls, drive throughs or traffic jams), so have fun.
