This subject is featured in the books 'The Severerd Alliance' and 'Morrissey's Manchester'.
The original band was Ed Banger and the Nosebleeds. Ed, the vocalist, then left and joined Slaughter and the Dogs, another Manchester punk band.
Morrissey then became singer of the new look Nosebleeds that included Billy Duffy (of Cult fame) on guitar. Duffy knew Johnny Marr from before and later brought Marr and Moz together.
The Nosebleeds, with Moz on vocals, only played live twice and then split. Duffy then also joined Slaughter and the Dogs and when Ed Banger quit, Morrissey went for an audition but was unsuccessful. All of this took place in 1978.
Hope that answers your question?