Particles accelerating in a cyclotron at sufficiently high energy reach relativistic speeds. You have to account for their relativistic mass increase to get the cyclotron to work. Figuring this out was a big issue in cyclotron design in the 1930s. The remedy is to strengthen the magnetic field near the outer edge of the cyclotron where particles move fastest, by adding coils there
to carry more current. I don't recall what the energy is where this becomes necessary - it is certainly needed at tens of MeV.
Plutonium was first synthesized in a cyclotron by Lawrence's group at Berkeley.
I don't know what energy they used so I don't know if they needed the extra coils, but they did know of the effect and must have considered it.
Also, U235 was separated at Oak Ridge using machines called Calutrons invented by Lawrence that might have encountered the same problem -- at least they must have considered it.
The atomic bomb certainly had a cursory dependence on special relativity. E = mc^2, you know.
Klystrons were used in WW2, and the beam current of a klystron scales as the beam voltage to the 3/2 power (due to space charge limits). Modern klystrons operate with relativistic electron beams, but I don't know if any of the WW2 ones did.
Is there any part of the theory or design or implementation of the atomic bomb that depends on E = m*c^2? Or is it: "if you could theoretically weigh the end-products, you would get a slightly smaller answer than the before-products". Seems like it would have been known that there was a lot of electrical potential energy stored in the nucleus (after the discovery of the proton).
Calculating the energy yield of a fission event is done by comparing the masses of the initial and final states. The difference is about 1/5th the mass of a nucleon.
Another situation that occurs in nuclear weapons and in reactors is the scattering of energetic photons off materials. This process is inherently relativistic when the photons have energies comparable to or greater than the rest energy of an electron. (Differential) cross section computations are necessarily relativistic.
At Los Alamos, various techniques using energetic photons and electrons were used to diagnose implosion systems in development of the plutonium bomb.
Plutonium was first synthesized in a cyclotron by Lawrence's group at Berkeley. I don't know what energy they used so I don't know if they needed the extra coils, but they did know of the effect and must have considered it.
Also, U235 was separated at Oak Ridge using machines called Calutrons invented by Lawrence that might have encountered the same problem -- at least they must have considered it.