Thanks, and sure. You're right to be skeptical. The renewability of uranium isn't a widely accepted idea and unfortunately is hard to definitively prove experimentally due to the hilariously long time scales. So we're left with less rigorous claims of this to treat rationally more than scientifically. Also, haha: 'fishy'!
Here's a good source: http://www.mdpi.com/2071-1050/4/11/3088/htm
And it points to an older one: http://tmtfree.hd.free.fr/albums/files/TMTisFree/Documents/E...
From the first:
"One additional aspect of nuclear sustainability—noted long-since by Bernard Cohen—is that a significant fraction of the nuclear fission energy resource is in fact completely “renewable” in the same sense as wind and solar energy [32]. Wind and rain constantly erode the Earth’s crust, which contains an average uranium concentration of 3 parts per million. Rivers then carry this dissolved uranium into the oceans, at a rate of approximately 10,000 MT per year [33]. In a breeder reactor energy system, this is a sufficient rate to supply the world’s entire electricity demand at the present time more than five times over—or is roughly one quarter of what’s needed to supply a continual 100 TW to a hypothetical global civilization of 10 billion persons which is energy supply-replete by any contemporary measure.
As the crust is being eroded by rivers, it is constantly replaced by new layers of rock being pushed upward by plate tectonic processes. The supply of uranium in the Earth’s crust is effectively inexhaustible, on the order of 40 trillion metric tonnes, a factor of 10,000 more than is present in the oceans. At present erosion rates, this source of uranium would last on the order of 4 billion years, similar to the timespan over which the Sun will become a red giant.
Therefore, this assured source of “continually mined-by-Nature and oceanically presented” uranium will last as long as life on Earth does—even if burned at rates sufficient to supply a large fraction of a fully-developed human civilization—and represents an astronomical amount of nuclear energy, one that is in fact truly renewable and inexhaustible by any human measures."
EDIT:
As for increasing concentrations of uranium in the sea, it's already in equilibrium so the incoming stuff that leaches in through rivers is leaching back into the rocks, giving us the equilibrium concentration we observe today. The point here is that if we start extracting it that concentration is not expected to drop because we can't offset the balance very much.