So your timeline is a little off and it is worth clarifying. Prior to the discovery of nuclear fission, uranium was not considered of extremely high value. Its main usefulness was either as a ceramic glaze or as something from which radium could be (laboriously) extracted from.

The discovery of nuclear fission (splitting of atoms) in 1938/1939 got many nations interested in uranium, as any country that had any ambitions of creating nuclear reactors or nuclear weapons would need huge amounts of it. So many nations began to seek out the ores that were in their boundaries and the ores outside of them.

Once the US Manhattan Project got going, acquisition of uranium ores was of the main parts of it. It got some of its ores from domestic sources (the US has some middling ore sources in the southwest, which were so middling that prior to this, the uranium component of the ores was typically considered a waste product), and some from Canada (who had somewhat better ores, but still middling). But the main area of interest was the Belgian Congo, which had the richest sources of uranium ore in the world. Belgium had been taken over by Nazi Germany, but much of the uranium was spirited out of German control just prior to that. The US reinvigorated work in the Congolese mines and was able to hoard tens of thousands of tons of raw uranium.

At the same time, the US also embarked on agreements with the British to secure a monopoly over other known uranium (and thorium) sources globally. So they hammered out agreements with Brazil, Portugal, and many other nations that had some kind of access to uranium ores. The goal here was to deny them to the USSR (or anyone else), and thus perpetuate an American nuclear monopoly. The USSR was very uranium poor after WWII (but they adapted to this, eventually), and, again, without uranium you can't have a nuclear program.

The US continued to be desperate for as much uranium as it could acquire through the early Cold War, because this directly impacted the size of its nuclear stockpile. It pursued a number of policies to this end, including artificially manipulating the domestic uranium price to encourage prospecting. By the 1970s however it had more uranium than it really knew what to do with and allowed it to become a more "normal" market.