Nothing in our technospehere is truly independent of the fossil fuel infrastructure and production. But this is not a grave problem since there will be expensive oil available in large quantities for a very long time even if very cheap oil in rediculously large quantities will be missed.

If another energy source produces much more energy in a usefull form like electricity then it needs lubricants, hydraulic fluids, diesel etc it can pay for very expensive hydrocarbons. It will more or less multiply the hydrocarbon input into a lot more electricity then burning the hydrocarbons would give.

Nuclear power, wind power, etc can then use heavier and scarces oils, then oils cracked with hydrogen made with electricity, oils syntehisized from coal, biological oils or oil synthetisized from raw biomass and hydrogen from electricity.

If no technological breakthrus are made I suspect that the latest generations of nuclear powerplants will be used for a lot longer then 45-60 years. You can not let the structure or systems deteriorate due to the security needed so minor subsystems are replaced as they wear down. This leaves large parts as the preassure vessel as the limiting parts while the rest of the systems are in good shape. They can be heat treated in place to release stresses in ther crystaline structure built up by irrradiation. The inner surface layer with most of the radioactivity can be etched away and then it is possible to work inside them and inspect and replace parts but they do wear out. I suspect that it then will make sense to even swap out reactor vessels. My guess is that the life lenght limiting part will be the concrete in the walls. I have no idea how long it will last, perhaps hundreds of years?

One point i find intresting is that a lot of the parts are metallic and can be melted and the metals reused, often directly and for some parts after a long period of underground storage to wait out the radioactivity. The largest volume of waste from the plant itself seems to be ion exchange filters and misc clothing and tools. It ought to be possible to reduce this waste stream with new materials that are washable but I am speculating about that. I get the impression that a nuclear technology that use breeder reactors can be sustanable more or less indefinately digging up the parts buried a few thousand years ago and remelting them along with fresh low grade ores for the next generation of powerplants.