In fact the word has existed since the 1840s, well before the atomic age gave it its present meaning. Nucleus in Latin was the kernel of a nux, or nut, and had come into English in the early 18th century as a technical term for the heart of a comet, before meaning the core of anything. (For a time it could serve as the adjective as well, so that a skeleton crew would be a nucleus crew, not quite what we mean by a nuclear crew now.) Nuclear likewise began as a technical term, mostly to do with the innermost part of cells - it wasn't one of those general words later taken up by scientists and borrowed back later still by laymen as metaphors, like paradigm say, though specialists have often borrowed it from each other, with Freudians writing about the nuclear complex, phoneticians about nuclear syllables and sociologists about the nuclear family.
Nuclear has spread itself since Hiroshima. If anyone mentioned "nuclear war" to Rutherford he would probably have visualised a nucleus being bombarded by neutrons. We, on the other hand, forget the atom's miniature systems, and imagine the larger-scale horrors.