Well, according to "Sustainable Energy Without the Hot Air"

>Japanese researchers have found a technique for extracting uranium from seawater at a cost of $100–300 per kilogram of uranium, in comparison with a current cost of about $20/kg for uranium from ore. Because uranium contains so much more energy per ton than traditional fuels, this 5-fold or 15-fold increase in the cost of uranium would have little effect on the cost of nuclear power: nuclear power’s price is dominated by the cost of power-station construction and decommissioning, not by the cost of the fuel. Even a price of $300/kg would increase the cost of nuclear energy by only about 0.3 p per kWh. The expense of uranium extraction could be reduced by combining it with another use of seawater – for example, power-station cooling.

https://www.withouthotair.com/c24/page_165.shtml

https://www.withouthotair.com/c24/page_174.shtml

This article estimates that the cost of seawater extraction would be roughly 10x the current market price, although not much word on the energy consumption.

[0] https://sci-hub.tw/https://doi.org/10.1016/j.pnucene.2017.04...

The sun also will burn out eventually, and every other star as well.

First section your Article: "As of 2017, identified uranium reserves recoverable at US$130/kg were 6.14 million tons (compared to 5.72 million tons in 2015). At the rate of consumption in 2017, these reserves are sufficient for slightly over 130 years of supply. The identified reserves as of 2017 recoverable at US$260/kg are 7.99 million tons (compared to 7.64 million tons in 2015).[9]"

Doesn't seem clearcut

No, these are the reserve that are available at a given price. It is very sensitive to technology. Moreover, price of uranium is a small part of the cost of a powerplant, so there is not as much price sensitivity. We're still good for around ~200 years.

Also, fuel can be reprocessed, and we can use other things than Uranium

Almost all predictions of running out of ores have been wrong. They’re based on “proven reserves,” but mining companies don’t bother proving much beyond 50 years worth.

I wouldn't worry too much about running out of nuclear fuel. There's also Pu used in MOX fuel, breeder reactors, nuclear fuel reprocessing.