The biggest problem with nuclear power isn't the finite quantity of obtainable uranium on the planet, it's that we are still building new reactors based on designs and technology that are 50 years old, and every new or clever idea anyone has come up with or even proved out is either completely ignored or outright repressed. Argonne National Laboratory came up with a completely different reactor design and fuel life-cycle over a decade ago before all their funding dropped out from under them. Even then, they'd already built the reactor and successfully proved they could make it work, and work well.
The basic idea was to combine a liquid-metal cooled reactor (ideally lead-bismuth eutectic) with a pyrometallurgical fuel reprocessing scheme. The liquid metal cooled reactor is inherently safer than the light-water reactors we still insist on using because it is capable of cooling itself passively without any pumps at all. (Ask Russia. They've been using them on their subs since the beginning of the Cold War.) Meltdowns wouldn't be impossible, but with proper reactor design they'd practically have to be deliberately instigated. This type of reactor is also capable of breeding and burning with ease the transuranics currently considered unusable (and highly dangerous) waste in the current reactor/fuel-cycle scheme.
As for the pyro-reprocessing:
# It separates out all actinides, and therefore produces fuel that is heavily spiked with heavy actinides, such as Plutonium (240+), and Curium 242. This does not prevent the fuel from being suitable for reactors, but it makes it hard to manipulate, steal, or make nuclear weapons from. This is generally considered a fairly desirable property. In contrast, the PUREX process can easily produce separated Uranium and Plutonium, and also tends to leave the remaining actinides (like Curium) behind, producing more dangerous nuclear waste.
# It is somewhat more efficient and considerably more compact than aqueous processing methods, allowing the possibility of on-site reprocessing of reactor wastes. This circumvents various transportation and security issues, allowing the reactor to simply store a small volume (perhaps a few percent of the original volume of the spent fuel) of fission product laced salt on site until decommissioning, when everything could be dealt with at once.
# Since pyrometallurgy recovers all the actinides, the remaining waste is not nearly as long lived as it would otherwise be. Most of the long term (past a couple hundred years) radioactivity produced by nuclear waste is produced by the actinides. These actinides can (mostly) be consumed by reactors as fuel, so extracting them from the waste and reinserting them into the reactor reduces the long term threat from the waste, and reduces the fuel needs of the reactor.
It won't by viable forever any more than burning fossil fuels is, but it would buy us more time to develop and implement truly renewable power sources. I don't think liquid metal reactors and pyro-reprocessing should be implemented to the exclusion of renewable technologies available now, but neither do I understand why everyone insists on ignoring it and all it can offer.
