I did a little more research on Thomas. As you know..he was aged 36 in 1881 census. Born in Endon, Staffordshire. Occ: Joiner. -=-= 1871 census. I found Thomasaged 27. Occ: Joiner.Single. Also present was his mother Ellen. 58. Widow. + Hannah Hassall..age 13 listed as grandaughter to Ellen. =-=- I looked for the family in the 1861 census to see who Ellen's husband name was However, I couldn't find them in 1861. =-=- I then looked for Thomas in the 1851 census.
I found Thomas aged 7 years born in Endon. Father Enoch Baddeley..Mother..Ellen. Enoch'soccupation= Victualler. However, there were several step children listed=Hassall's. It seems Ellen was married before.
I looked for a Enoch Baddeley marrying a Ellen. I found marriage ref. #Enoch Baddeley married Ellen Palmer HASSALL in 1842. Marriage reg'd in Leek, Staffordshire. Endon and Tittesworth come under the Reg'd Disc't of Leek. #These are your Thomas's parents. =-=- I found Ellen HASSALL in the 1841 census. She was a Widow. Married the following year to Enoch..1842. -==- =-=-=-
I believe I may have found Emily and daur Gertrude in Providence, in the 1900 census.
I found her in the 1910 and 1920 census in Lord h'hold.
Just trying to help.
p.s. I found only one ref for a Mabel Ann Baddeley born in Staffordshire,appx the time your Mabel Ann was born. This Mabel Ann was reg'd in the 1st qtr of 1878 in Leek. There were no other Mabel Ann BADDELEY's reg'd before 1878 or after