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: 
In Brazil "pure" ethanol (94% ethanol, 6% water) is sold at every gas station, and pretty much every new car sold can run with any proportion gasoline-ethanol mixture (they call it "flex" engines).

So, the infrastructure for a fuel that can power a combustion engine for a heavier than air aircraft does exists independently from oil since the 1970's.

The history would certainly be different, and possibly there would not be the political-economical incentives for airplanes to be what they are today, but it is quite possible they would be invented and be used for local travel in places where sugarcane is viable.
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.