> There were engineering questions around how to purify uranium and how to actually construct the weapon etc. But the phenomenon was known.
FWIW, constructing a weapon with highly enriched uranium is, relatively, simple. At the time, the choice was made to use a gun-type weapon that shot a projectile of highly enriched uranium into a a "target" of highly enriched uranium. The scientists were so sure it would work that the design didn't necessitate a live test. This was "little boy", which was eventually dropped on Hiroshima.
Fat Man utilized plutonium which required an implosion to compress the fissile material that would set off the chain reaction. This is a much more complex undertaking, but it's much more efficient. Namely, you need much less fissile material, and more of that fissile material is able to participate in the chain reaction. This design is what allows for nuclear tipped missiles. The same principles can be applied to a U-235 based weapon as well.
The implosion based design is super interesting to read about. One memorable aspect is that the designers realized that applying a tamper of uranium (U-238) around the fissile material allows for significant improvement in yield. The chain reaction is exponential, so the few extra nanoseconds that the uranium keeps the fissile material together leads to significant increase in yield.