We would absolutely be studying them intensively at the time
This argument is just from human nature. Given that the Cold War was well underway in the late 40s, and a space race was about to begin, I can't imagine any other outcome from discovery of other civilizations in the Solar System then large-scale jockeying for diplomatic and scientific alliances between the West and the Soviet Bloc.
Given that, it is almost assured that both sides of the Earth divide would be heavily invested in looking at what was happening on Mars.
There was plenty of excellent detection equipment available.
The Arecibo Observatory was complete in 1963. The Deep Space Network started in 1958, and by the late 70s there was nearly full planet coverage in L- and S- band (on the border between radio and microwave). Optical range telescopes had already been numerous for centuries.
There were also plenty of in-space satellite telescopes by the 1970s. Cos-B was a European gamma telescope launched in 1975; there were two HEAO x-ray telescopes launched in 1977 and 1979; several UV telescopes including the Soviet Union's Orion series launched in 1971 and 1973; etc.
A nuclear blast is easily observed from the distance of Mars
Let us assume that a 1 MT weapon yields 1% of its output energy as prompt gammas, using rough estimates from Wikipedia. That means 4.18e13 J is released just as prompt gamma energy. Given that Mars is 225 million km away from Earth, projecting that energy onto a sphere of that distance gives an energy concentration of 6.6e-11 J per m$^2$. Translating that into ergs / cm$^2$ gives 6.6e-8 ergs. While that isn't a lot, that is equivalent to the lower end of observed gamma ray bursts.
This is just back of the napkin math, but the power of a nuclear weapons indicates that the gamma burst at least should be clearly visible from Earth, and there were satellites in orbit such as the Vela series starting in 1963 that could detect such a burst.
It is very likely that because of the attention we would be paying to this nearby Martian civilization, and because of the high energy gamma burst, the space agencies of Earth would know almost immediately if nuclear weapons were used on Mars.