I cannot find either Wm, Mary or Fredrick in 1881 only the Eliza you found in the workhouse.
The family in Wales in 1891 and 1901 is not the same family as Wm Hathaway b 1836 the Saddler. , I found this family in 1871 a Wm Hathaway 35 a Tin Roller, wife Mary 34 various children including a JOSEPH aged 2 months in Walsall.
I don't think the marriage that was found of Elizabeth Hathaway is yours either as this family also had a daughter Elizabeth so it may be her.