>did we really need quantum mechanics to build an atom bomb?

Yes. Nuclear reactions require understanding and modeling of the strong force, you can't understand or even see what protons and neutrons are without understanding the strong force. The mixture of positively charged and neutral particles being stuck together with enormous force which essentially does not exist at all outside of the nucleus of an atom. (there is more than three pounds of force between every pair of protons inside every nucleus with the strong force counteracting the electrostatic force)

You couldn't design a bomb without being able to model the strong force and you couldn't get to that point of investigating the atom without coming up with QM.

You couldn't get the idea of isotopes and enriching U-235 to U-238 or transmuting uranium to plutonium without understanding QM.

Or the circumstances that would lead someone to blindly creating a controlled nuclear reaction without coming up with QM in the process would be pretty absurd.

The idea for the bomb came from the understanding of the strong force. Step one: notice that there's a crazy powerful force keeping positively charged particles stuck together in the nucleus. Step two: the eureka moment of realizing you can "release" that force by causing a chain reaction of fission in heavy elements.