Why don't aircraft use nuclear propulsion?
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.