Question: In a world without fossil fuels, would air travel ever really be developed?

Without fossil fuels, the most likely path that could lead to the development of a proper industrial revolution would eventually be nuclear power and renewables. This world would probably start with hydro-electric power, in which they then move on to nuclear as scientists living in the cities that build up around dams are able to discover more about nuclear physics that would allow this to be harnessed as a fuel source.

While there are other questions that could be raised, how would air travel be developed in this sort of world? Could airships remain viable in the face of electrified railroads and nuclear transport across oceans?

Would fixed wing aircraft or helicopters eventually be developed? What fuel sources might they use?

Answer: 
### We were flying for [hundreds of years](https://en.wikipedia.org/wiki/Early_flying_machines) before putting an engine on planes:

* We had man-carrying kites going back to possibly the 7th century.
* 1783 we had manned hydrogen balloon flight.
* First heavier than air craft (no human on board) was 1655 apparently ([spring powered ormithopter](https://en.wikipedia.org/wiki/Early_flying_machines#The_17th_and_18th_centuries)).
* In 1853, British engineer George Cayley built the world's first real glider.

With no petrol, the wright brothers plane as built couldn't exist (they used gasoline), but we had batteries and electric motors at this point in history, they have a lower power to weight ratio, but that first flight could've occurred electrically in the same period.

Long distance air travel would be harder, batteries are heavier than the same amount of fuel, but lacking fossil fuels, we can still refine ethanol from biomass. There are [ethanol powered aircraft](https://en.wikipedia.org/wiki/Embraer_EMB_202_Ipanema), which are actually cheaper to run than gasoline ones. That linked one is commonly used for crop dusting, and can be powered by ethanol generated free from waste parts of crops. There's no reason that can't scale up to a small airliner-sized aircraft.
Piston and jet powered flight could still be developed without reliance on fossil fuels. We use fossil fuels for gasoline, diesel, and jet fuel because it's relatively easy to prepare from the oil we extract from the ground, the fuels are energy-dense and the engines are power-dense. However, both piston and jet engines can run on a range of fuels that could be synthesized or refined from sources other than petroleum. Jet fuel can be refined from vegetable oil. For piston-powered flight, you need something that behaves pretty much like gasoline in terms of energy density, combustibility, and storability - a blend of mostly alkanes in the right range (heptane, octane). It is possible (not necessarily easy or efficient) to synthesize them from hydrocarbon sources - vegetable oils and/or other plant matter. If you have cheap and abundant nuclear power, that can help to make synthesis of aviation fuels economic.

It would be far more plausible to see aircraft powered by liquid fueled piston or jet engines than compact nuclear reactors, even if the fuels have to be synthesized because nuclear power production scales up well, but doesn't scale down well, and mobile reactors bring safety issues - collisions/crashes can trigger dangerous leaks of radioactive materials.