Nuclear propulsion research for aircraft was abandoned during the 50's.
Why wasn't it revived ever?
TL;dr - too heavy :-)
It's just not a good energy source for something like an aircraft.
Nuclear energy is superb for instances where you need continuous output over a long period of time, for example a satellite, which is going to be there for years without maintenance or refuelling. A very small amount of nuclear material in an RTG such as those on Voyager 1 and 2 can provide heat energy that can be used to power the satellite. This model works in this use case because the only real alternative (solar power) hasn't anywhere near the output required as the Voyager craft fly ever further away from the Sun.
Also, submarines are a good use case - if you remain under the sea for months or even years at a time, diesel or anything that uses oxygen is not suitable, so a nuclear power plant is ideal.
But while for a satellite the power unit can be very small as the actual output power required is small, to power an aeroplane or a submarine you require a large power plant, and that will also require a lot of shielding (as a side effect of nuclear reaction is radiation...)
On a submarine, that's okay - weight can be dealt with, but on an aeroplane weight is key. You just can't do it efficiently and safely. You need shielding that can cope with a crash - which is going to be massively heavy, so you won't have any passenger or cargo payload left.
To a large extent, it wasn't revived because the only realistic use case became obsolete. The USA and USSR were both interested in nuclear-powered long-range bombers. The plan was to have a fleet of bombers loitering in the Arctic so that, if nuclear armageddon was required, they'd already be half-way to their target. They'd also be very hard to destroy as part of a first strike, since they'd be out of range of enemy fighters and widely dispersed. A nuclear-powered plane wouldn't need to refuel so the limitation on mission length would be basically crew resilience. With relief crew on board, you can easily imagine a mission lasting many days. All of this would give a credible nuclear deterrence.
However, the advent of high-altitude surface-to-air missiles in the late 1950s meant that any first or second strike with bombers would be vulnerable to the enemy's air defences. As a result, missiles became the preferred delivery method for both first- and second-strike capabilities. Survival against a first strike was now ensured by having widely dispersed silos on land, and missiles on submarines.
Reconnaissance is the only other application I can think of for a plane that can be aloft for multiple days at a time. But that's still vulnerable to anti-aircraft defences and the recon mission was also moving towards another platform (satellites) from the late 1950s.
For anything where you don't need huge endurance, nuclear power doesn't make a lot of sense. The two main problems are the inevitability of crashes and weight. The B-36 that was modified as a testbed had a 12-ton shielded cockpit module and at least five tons of lead between that and the reactor, along with tanks of water that acted as both shielding and cooling. All of that is heavy (and needs to be close to the engines) so it needs to be in the middle of the plane. So, if you wanted to build a nuclear-powered airliner (e.g., for non-stop Europe–Australia/New Zealand flights), you'd need another shield behind the reactor and the reactor and shielding would seriously decrease the size of the passenger cabin and, hence, potential revenue.
On the plus side, you'd save carrying 150 tons of kerosene on your long-haul flight but it seems that nuclear is still heavier overall. For example, there were plans to build a 15,000ft runway at Carswell AFB to allow the proposed Convair X-6 to take off. For comparison, even at large commercial airports, the longest runways tend to be in the 10-13,000ft range.
Because there was no practical purpose such an aircraft would've served. First and most important, the safety concerns over such an aircraft design would make its use in civil aviation more or less impossible, especially when more and more countries are shying away from nuclear power.
This leaves the military. There are quite a few issues with such an application. The most important of these would be the weight. The cost in weight for shielding the aircrew and the weapons (which by default would have to be nuclear), would be prohibitive. For example, the first (and only) 'nuclear' aircraft , the Convair NB-36H had a 11 ton shielded cockpit.
Special shielded cockpit being installed in the NB-36H; image from aviation-history.com
That's about a third of the aircraft payload. Add this to the reactor shielding, and the aircraft would not have any meaningful payload. Even in case of ships, the nuclear powered ones come with signinficant weight penalty, saved only by the low fuel cost (and space) associated with it.
There is a significant cost associated with maintaining nuclear weapons and safely maintaining them, which would be prohibitive for a fleet of nuclear powered aircraft. USAF sunk nearly a billion dollars in the nuclear aircraft without anything to show for it. Nuclear tipped missiles are more cost effective and more survivable that these lumbering beasts in air.
The only virtue of a nuclear aircraft is its virtually unlimited range and endurance , which is useful only in case of strategic bombers on continuous patrol with massive free-fall nuclear weapons, which has no meaning today (ironically due to nuclear submarines with missiles). Advances in aircraft and propulsion has resulted in a number of aircraft with >10,000 range, which is more than enough for all practical purposes and can be extended through air-to-air refueling.
Even is all these technological issues are overcome, nuclear powered aircraft would be a overkill- there is no point in having an aircraft of unlimited range and endurance is the the crew can't eat- even in case of nuclear submarines, the food is the critical resource. Nuclear propulsion maybe used in an interstellar craft which requires fuel supply for very long with severe limitations in fuel mass and volume, but in case of an aircraft, it would be of no use.
What if the nuclear powered aircraft crashes? It would be near impossible to design a reactor that could withstand a 500+ mph impact, and you'd have a serious radiation mess to try to clean up.
The 'direct cycle' engine, where air is heated directly by the reactor, irradiates the air and leaves a radioactive trail behind it. Both the Soviet TU95LAL and the US SLAM nuclear cruise missile had plans for a direct cycle engine. With the SLAM, the radioactive trail was considered part of the weaponry. Of course, neither side fully explained how they could protect their people around the area where such a monstrosity would be launched.
The Convair NB36H had plans for an indirect cycle engine, which didn't leave a radioactive trail, but it never progressed beyond the stage of hauling an operational reactor aloft.
In the end, the perfection of aerial refueling in the 1950's obsoleted the need for the extra range that a nuclear powered aircraft promised, while the complexities and potential dangers were never fully resolved.
Already great answers here but I`d also like to add...
In the 50's it was the atomic age. We thought splitting the atom was the best thing since sliced bread. But that was at a time when we, as humans, had no idea of the long term effects of radiation and radiation poisoning. It really was not until a decade had passed since the dropping of the bombs in WWII that the long term effects became apparent and were simply horrible. In the mean time we had done all sorts of above ground testing with "volunteers" in the trenches exposed to blast radiation and fallout.
Until we woke up to that, all sorts of fanciful ideas came up that today we would say are ludicrous. Among those was the notion of putting a reactor inside an aircraft and flying it over populated areas.
Today we know better, and the probability of anyone doing such a thing, except perhaps for some remote drone to fly way out over the ocean perhaps, is zero to none. At least in developed nations.
There were other weird ideas at the time too. Microwave heating was discovered in that era too. Someone thought we could heat our homes with microwaves instead of furnaces. The house would be cold, but our bodies would be toasty warm.
The point is when new technology comes alone, folks usually try to use it in novel and different ways. It isn't till later that a sanity check kicks in.
The existing answers have already covered the weight, lack of need, and crash safety concerns well, but there is also an additional reason, especially for civilian airliners: hijacking. Securing a nuclear reactor at a power station is feasible because it sits in one place and we can build fences and walls around it with armed security. While submarines and aircraft carriers are not stationary, anyone that tries to attack one to get its fissile material is going to have a very bad day on the wrong end of a naval gun (or torpedo, missile, etc.)
Unfortunately, history has shown us that hijacking an airliner is much easier than attacking a carrier battle group or a nuclear power station. If we started putting significant amounts of fissile material onboard airliners, rouge countries or terrorist groups seeking such material would have more incentive to hijack airliners in order to acquire the fuel. Considering the remote corners of the world that airliners must visit, both for cargo and for passengers, it wouldn't be very difficult to hijack several of them in short order or possibly even steal the fuel from them while they sat on some remote ramp. Relative to the other ways significant amounts of fissile material can be acquired, this would be pretty straightforward.
Additionally, if someone tried another September 11-style attack, the airplane would now be a giant dirty bomb. Not good.
Along with all the other reasons, there's another - nuclear reactors contain a lot of energy but are not so good at power.
An aircraft takes off using 100% throttle. After reaching cruise it throttles back to somewhere around 55 to 70%. During a war mission, they would throttle back up to 100%, basically instantly.
Nuclear reactors do not like to throttle. The one near my home can throttle about 15% over a 24-hour period. It is possible to make ones that are better than that, like on nuclear submarines, but you start getting more and more complex.
So that's where they were at in the 50s when the Atlas was beginning to mature. Why bother with all that complexity when you can have the same bomb just sitting in a silo ready to go when you push a button?
I suppose Nuclear powered airplanes never went from experiment because of the fear one of it falls on your backyard. A Steam-powered airplane went fully airborne, the Besler Steam; the possibility of a Nuclear Reactor driving propellers as in B-36, but through Steam Turbines, seems realistic. A similar concept was used for interplanetary probes, relying on heat from radioactive isotopes to produce energy. USSR was attributed considering a Nuclear Energy powered bomber, the USA attempted purchasing the Saunders-Roe Princess for converting it to Nuclear Power, but the Flying Boat was severely corroded from lack of money to properly preserve it.
We can discuss this on a very general level, without diving into technical details.
The increasingly demanding frame conditions in the succession "land — sea — air" apply not only to the reactor but also to the vehicle proper. On land and on sea loss of propulsion or some structural damage is usually not fatal (after all, a land based nuclear power plant is stationary by design); in the air it often is.
It is no coincidence that for example spent fuel containers are transported by train and truck, not flown around, even though that would avoid a lot of trouble with protesters.