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.
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.