Not directly related to the fuel used in the aircraft, coal is also used to create coke which is used in the creation of metal products
In 1709, Abraham Darby I established a coke-fired blast furnace to produce cast iron. Coke's superior crushing strength allowed blast furnaces to become taller and larger. The ensuing availability of inexpensive iron was one of the factors leading to the Industrial Revolution. Before this time, iron-making used large quantities of charcoal, produced by burning wood.
As other answers show fossil fuels can be replaced with other fuels. However the lack of easily and cheaply available fossil fuels would impact the course of the Industrial Revolution in your world impacting the availability of many technologies.
Additionally the lack of fossil fuels means that the use of wood as a fuel remains high leading to higher levels of deforestation which can deeply impact a region's ecosystem. Japan managed to prevent deforestation by changing their use of forests (source: https://www.appropedia.org/Japan_Forestry). Easter Island, however, did not.
With the loss of their forest, the quality of life for Islanders plummeted. Streams and drinking water supplies dried up. Crop yields declined as wind, rain, and sunlight eroded topsoils. Fires became a luxury since no wood could be found on the island
Would Britain have avoided Easter Island's fate without fossil fuels to burn for simple things such as heat and cooking? A different approach to the use of wood and forest management would be required. Further impacting the course of industrialisation. Farming is also impacted by industrialisation, think of all the tools and machines. Further farming depends on fossil fuels for synthetic fertilizers (https://en.wikipedia.org/wiki/Fertilizer#Nitrogen_fertilizers).
Considering the reduced farming efficiency means that large part of the population is involved in food production so there are less engineers and scientists for developing alternative methods for creating cheap metals, electricity generation, or even nuclear technology. The farming efficiency will also impact the availability of bio fuels.
All that being said the Industrial Revolution would probably still be possible without fossil fuels but it's course would definitely be different. I can imagine that societies would try to find faster growing trees to supply their energy needs, maybe even bamboo which can also be turned into charcoal. Probably different furnace designs would generate cheap metals with charcoal or bio fuels. Electricity will likely start to play a role in metal smelting and refining earlier on as well, see eletric arc furnances for example. Once this thin edge of industrialisation's wedge is in the developments will continue. However the industrialisation speed and scale would be impacted by the need to constantly balance the environmental aspects to maintain healthy forests and farmland. Failing to do so would hurt a society without fossil fuels a lot quicker stalling their industrialisation.
As laid out by others aircraft would be possible without fossil fuels. However the total energy available is simply more limited without fossil fuels. Most technologies, including railroads and flying would be less wide spread and thus available only to the rich elite of society.
Hydropower is also a lot less available especially before Hydropower dams can be built, still being based on waterwheels. Wind power would be implemented more since windmills can be placed in more locations. Batteries would be a lot more important in this world dependent on electricity, as we start to see in the world today. Battery powered aircraft are possible but require more energy dense batteries than we have today. Considering the importance of batteries in your world these would surely be invented.
I expect nuclear power to be a bigger game changer than in our world, making power available more cheaply at a bigger scale. Either that or wind and eventually solar power would play a bigger role.
As for your question "Could airships remain viable in the face of electrified railroads and nuclear transport across oceans?" Probably yes, aircraft would still be faster than ships and trains generally can not cross oceans and seas. Also aircraft need less infrastructure than trains which I can imagine will be an even bigger advantage in a world where it less easy to mass produce many thing for a long time. Also don't forget the military applications that are unique to aircraft which will aid their development.