I haven't read Williams' play in years, but if I remember it correctly, the song is probably about Laura Wingfield, the younger sister in the play, rather than Woolf's essay. Her older brother, Tom, is nicknamed "Shakespeare", and there is a rather suffocating mother.
Generally speaking, Morrissey would be attracted to Williams more than Woolf. I don't think there's much in Woolf that would appeal to him, aside from her central interest in women's self-realization, or whatever one may call it. Williams, on the other hand, is of a piece with Truman Capote.