OK, this is an area where I have some expertise ( I work in a nuclear power plant):
If you visit Idaho you can see the two nuclear reactors that were built as test engines for possible US Military applications along with the first nuclear electrical generation power plant.
The project was stopped for 4 basic reasons: Shielding, weight, radiological contamination from a crash, and cost.
While reactor design has improved a lot since then you are not going to eliminate the weight and crash concerns.
However, the ideal nuclear system can indeed power a "jet" type engine (Brayton Cycle). The compressor sends compressed air to the reactor core, where the air is heated to potentially several thousand degrees, and then the hot air is exhausted through the nozzle after shedding some energy in the power turbine needed to turn the compressor.
Control and durability of such a reactor to allow it to rapidly ramp up and down are hurdles that are currently unsolved even if the weight and crash issues did not exist.
INL (the Idaho DOE nuclear development and test laboratory, where the aircraft engine test reactors are) is working on hot gas reactor designs right now for a high efficiency Brayton cycle engine which is aimed more at electrical power generation than anything (and the pebble bed reactor is a lessor efficient Brayton cycle engine design). I am unsure of the funding for construction of a test reactor - and I suspect it will be 10 to 15 years before even a test reactor is built.
Key in nuclear reactor design is that it is not currently possible to operate a reactor at the temperatures needed for a brayton cycle unless the reactor is very massive - just from a structural standpoint. More mass is needed for radiological shielding; and more yet if the reactor is designed to be contained in a criticality accident (to keep the zoomies contained and not spread all over the countryside).
A Steam Cycle reactor used for power plants also requires very massive structures for support and for containment (and the US did a very wise thing by requiring containment building just in case the "improbable" went wrong).
The world (and US) is just getting ready to start construction of the 3rd generation nuclear power plant designs - which will be much safer than previous generations of plants, I have read the papers and seen the preliminary 4th generation designs (expected commercialization timetable 25+ years from now), and also the summary papers on the 5th generation design hurdles that will take considerable material research in order to build (expected commercialization assuming materials are found would be 50+ years from now).
None of those designs will ever be weight efficient for use on a conventional aircraft, even if you could ensure that their would not be a criticality event or a crash.
On the fusion side of things.... I helped build portions of one of the latest fusion plasma experiments built in the US (is this machine cool or what... http://www.hsx.wisc.edu/
), and helped build a vessel to research the flow of cooling fluids for a fusion reactor (sorry - no cool pictures).
Any commercial fusion reactor is at least 50 years out; and they are also massive machines and require some degree of radiation shielding as well (not as much as a fission reactor). Due to what is known about the potential shapes a fusion reactor will require to be cooled by liquid metals (liquid sodium is the best bet)- which require a very massive system. By the time you add it all up this system is just as massive as a fission reactor; and I do not believe that anything that is currently conceived of as a real fusion reactor will ever be light enough for use on a conventional aircraft.
Of course, someone may well come up with a different idea on how to do fusion in the future. Figure at least decades from any idea to any commercially available system (the HXS device was at least 20 years from idea to first successful operation).
As others pointed out - the nuclear reactors used in spacecraft (and certain science or spy applications) do not produce much power at all and are not suitable for use in aircraft. Their are two forms of these reactors: Thermal Decay Heat devices that produce minimal electrical current via a thermocouple device, and higher temperature nuclear reaction devices that are not started until after they reach outer space (or are placed on the the ocean floor or a mountain top). These nuclear reaction devices can produce a fair amount of power by the use of a lot more and higher temperature thermocouple devices; and despite the fact that they do not have any degree of shielding they are in fact quite heavy for the power they produce . Both kinds of units can produce suitable electrical power for 20+ years (The Voyager Satellites launched I believe in the late 70's are still transmitting usable data to earth because of their nuclear "batteries"). I do not believe that nuclear reaction devices are currently being planned to be built - I do believe that a few were built.
so what can we do - and what does the future really hold:
Hydrocarbon based fuels will continue to be the fuel of aircraft for probably the next 100 years unless something really unknown comes along. Hydrogen will not be directly used due to the problems storing and handling it. However, hydrogen can be easily converted to hydrocarbons as needed. Thus the concept of a hydrogen powered future - with hydrocarbon jet fuel being produced for aircraft is a very viable future as we currently have the technology needed to build it.
Hydrogen will be produced first by electrolysis of water using power generated by nuclear power plants. This has already been costed out as part of the NP
2010 study: http://www.ne.doe.gov/np2010/neNP2010a.html
INL is working to develop the material technologies needed for a high temperature gas cooled nuclear reactor that will make hydrogen from water via high temperature chemical process that will be much more efficient than electrolysis. Currently known materials cannot withstand the required temperatures; but are close enough that it is felt that we should have suitable materials to test in a decade or so.
The US currently has the technical capability to start construction on a Hydrogen based infrastructure. The petrochemical industry already has Hydrogen pipelines that supply some of the oil refineries (current hydrogen production is from natural gas). If the US were to start down a path of building 10 nuclear reactors a year for hydrogen and electrical production the US could be oil import free in about 50 years and retire most coal fired generation (assuming expanding our current energy usage at recent historical rates). This would also assume a substantial amount of wind and solar generation (but wind and solar will never be economical to provide our base-load energy needs).
Of course, someone might wake up tomorrow - or sometime in the future - with an exciting idea on how to generate usable energy by a totally unknown process today. If they do - then that may well allow the flying cars of the future...
I wish you all the best.