"How Soon Is Now?" may be a 1980s classic, but musically, the song was influenced by the sounds of the 1950s and 1970s.
Johnny Marr wrote the song's distinctive riff before going into the studio. He was partially inspired by the "stomp" of 1970s disco. "As a kid I was fascinated by Hamilton Bohannon's 'Disco Stomp' and 'New York Groove' by Hello, and I wanted to make something with that stomp" (source), he said. But Marr was also passing through a Bo Diddley phase, also saying, "I was playing Bo Diddley stuff everywhere I went. I wanted it to be really, really tense and swampy, all at the same time." (Source)
The result was a disco "stompin' groove" coupled to a Bo Diddley-like tremolo.
Producer John Porter immediately recognized the potential within the riff, but he thought the chord progressions needed work, so he told Marr to jam on an old Elvis tune, "That’s All Right." After dropping the whole thing several keys, Porter and Marr tied the riff to the new chords and sent a tape to Morrissey. After piecing together lyrics from several songs in process, Morrissey recorded the vocals (allegedly, it only took him two takes to nail it).
Porter realized he had a distinctive package. "I thought...now we've got something that we can sell in America," he said. "Now we've got a band that could be like R.E.M. are now." (Source)