Your basic premise that we are in some way dependent on oil is incorrect. It's just the fuel of choice at the moment, an arrangement of convenience. It would be pretty straightforward to synthesize fuels and chemical feedstocks directly using power from nuclear plants. The only reason we don't do it right now is that it is not cost effective. Well that and the drawbacks to nuclear power. If the Germans could synthesize fuels in 1945...
Germans were converting coal to petroleum. Coal is also a finite resource. Nobody so far has come up with some synthetic fuel as potent as oil in terms to energy on return. You can convert your food to fuel for your gas tank, but that also requires fossil fuels for production (all the machinery to mass produce the corn which you convert to usable liquid fuel).
So, you could try to operate the present industrial infrastructure on ethanol and nuclear energy, but you won't have the same energy on return as you do with fossil fuels. Fossil fuels are concentrated solar energy and there is nothing else like them. As I pointed out, nuclear has a hidden subsidy of fossil fuels, likewise with synthetic fuels.
We will need to make do with less energy, but our civilization has come to demand vast more amounts of energy that we can possibly produce. Social and technological complexity requires that energy, and if that energy fails to be made available the system unravels.
This is what happened with Rome incidentally. In the 3rd century there are writers lamenting how harvests were decreasing in quantity. This was of course soil depletion. In the Roman civilization the main source of energy was solar energy gained through crops and exploited through animal and human muscle. Over time this meant there was less energy available to support a vast system which required it. This contributed to the currency inflation the later empire was continually plagued with. The Roman state had earlier enjoyed a vast energy subsidy in the form of conquered lands and slaves starting especially under Augustus, but by the 3rd century there wasn't many more peoples they could conquer or land they could exploit. This meant they reached the limits to growth and any further expansion would have been into lands which just weren't worth it at the time (Germania or the Sahara for example). So, they struggled with such problems and it eventually brought Rome to an end.
There are many similar parallels with our current industrial civilization. Instead of relying chiefly on solar energy gained through the soil, we are using oil instead. However, oil is a finite resource and we have reached peak oil already, which curiously corresponds with our economic problems. The same problem occurred in the US and UK in the 80s, though they fixed it for a time by opening up new drilling sites in Alaska and the North Sea. It got the economies back on track for a time. However, now we are have hit global conventional oil production. It will not increase anymore and we can expect it to start decreasing within this decade. The International Energy Association already quietly admitted we've reached peak oil a few years ago.
So, we've reached the limits to growth and now our currencies are clearly being inflated. Like in Rome, we debase our currency to prompt economic activity by borrowing from the future which cannot protest our actions in the present. When the secondary economy of goods and services cannot grow anymore, our financial industry started growing exponentially. Now for all the "money" that is produced in a single year it reflects less than a fraction of actual goods and services in the real world economy. There is vast amounts of hallucinated wealth in the system, which is quite similar to what happened in the later Roman Empire.
As Tainter explains in his work The Collapse of Civilizations
with such increased strain on a system the powers that be spend more resources on policing and legitimization of the status quo. This is exactly what is happening in the western world and much of elsewhere. Our rights are being dissolved, though at the same time the bread and circuses keep most people from really caring. This of course solves the problem for a time, though such initiatives require additional resources and put more long-term strain on the system as a whole. Hence it will come to a point where the whole system is unsustainable and just unravels.
When that unravelling happens it is unlikely there will be the means to operate industrial civilization as we currently have it at present. Like Rome, it will happen over time (it has already started by some estimates).
It won't be the end of the world, though in all likelihood we won't have the same standards of living as we enjoy now. A lot of social complexity will be lost, and with it knowledge traditions. Science is likely to take a hit given that it is supported by commercial and state interests, though when those interests are rendered powerless they won't be in a position to support anything. Religion gets away from this because it depends on ordinary people for its support base.
So, with both economic, social and scientific foundations undone, will nuclear energy be feasible? Unlikely.
The end of the world has been prophesied pretty much every year since Jesus was around. Time after time it has been proven wrong, and time after time new prophets have arisen. Always they claim we face circumstances we have never faced before, and each time we find a way through. Always the gloomy extrapolations are produced, and always they are invalidated by simply doing things differently.
I am not proposing it is the end of the world. I'm proposing the end to industrial civilization. Civilizations are impermanent and ours is no exception. Many like to think we are exceptional given our advanced technology, but such exceptionalism is ridiculous when you examine history.