There was the nuclear protest movement...
it was most active when there was greatest concern over nuclear weapons
1958 - Campaign for Nuclear Disarmament starts in Britain - it called for Britain to abandon its nuclear arms - failed because France and China were building stockpiles of nuclear arms
1982 - half a million people protest in New York over Reagan reigniting the arms race
Generally in America popular support far outweighed protest. Protest was only evident during the Vietnam war
In the USSR there was too little freedom to organise protest
it is possible that popular protest combined with scientific concern to influence the signing of the Partial Test Ban Treaty in 1963.
That might help. I agree though there's very little on it.