If nuclear war were to occur, would radiation linger in rain? I know contaminants get filtered out through rain, but does radiation?

Yes , radiation does get filtered.

Explanation:

Short-wavelength UVC is the most damaging type of UV radiation. However, it is completely filtered by the atmosphere and does not reach the earth's surface. Medium-wavelength UVB is very biologically active but cannot penetrate beyond the superficial skin layers. Most solar UVB is filtered by the atmosphere.