They pursued both routes to the bomb at the same time because they didn't know which one would work, if either. They had trouble with both (uranium enrichment proved harder than they thought; plutonium proved trickier to use in a bomb than they thought), but managed (by sheer luck) to have enough material from both process to be ready in mid-July 1945 to ship two bombs to use against Japan, plus one more plutonium bomb to test (necessary for the aforementioned trickiness).

After the Trinity test they had a pretty good idea of what the yields of the bombs would be. (They chose burst heights which suggest they were being conservative about the yield — they would have been good heights for 7 kiloton blasts — but they were expecting something in the 10-20 kiloton range for both.) That the two bombs had roughly the same yield is another coincidence; they originally had planned for the Little Boy bomb to have a yield in the 10-15 kiloton range and expected the Fat Man bomb would likely be lower, but it turned out that implosion worked better than they had expected.

As for why they used both, this is partially a much longer question of "Why Nagasaki?", partially a question of why they didn't split the very inefficient Little Boy bomb up into half-a-dozen Fat Man style bombs that used HEU. (I explore both in that blog post.)

As for the Cold War importance, it's interesting. The Hanford reactors gave them huge problems in the immediate Cold War and had to be disabled/shut down for awhile, cutting off their plutonium supply. This means that the US was for awhile dependent on its high-enriched uranium facilities at Oak Ridge, which were going great guns immediately after the war (when gaseous diffusion really got working correctly). But they had never used HEU in an implosion bomb before and didn't really want to use gun-type bombs. This created a lot of hassle for them until they finally tested a composite core bomb (an implosion bomb that uses both HEU and plutonium) in 1948. This is one of the many things which had the result of the US having a very small and fairly non-functional nuclear stockpile for the first few years of the Cold War.