A couple passages from JM's memoir suggests it's more him writing independently, with Morrissey coming in later:
New songs usually started with me recording the music on to a cassette and then giving it to Morrissey to write his lyrics and vocal lines, which he would complete within a day or two. Other times we would get together at my place and sit face to face, about three feet away from each other, while I played my new tune into a tape recorder that was balanced between my knees.
As I went around and around, the tune started to get psychedelic in my headphones and I knew I was on to something. I programmed a simple beat on the “drum machine and recorded the hypnotic rhythm guitar, and then came up with a two-note phrase that I put on top. What I’d done was nothing like the other two songs, and nothing like anything the band had done before either. When Angie got back I played her the demo and she thought it was great. Then I took the cassette of the three songs round to Morrissey’s, having written ‘Fast’, ‘Irish Waltz’ and ‘Swampy’ on it. He worked on it for a few days, and when he’d finished the lyrics the songs became ‘William, It Was Really Nothing’, ‘Please, Please, Please, Let Me Get What I Want’ and ‘How Soon Is Now?’.
He does say, however, that with some of their earliest songs he received words from Morrissey and worked from there.
"I used to bullseye womp rats in my T-16 back in Whittier, they're not much bigger than two meters.'" - Richard Nixon, Checkers Speech, abandoned early draft