Thanks for your replies. Yes, William and Thomas survived, they lived with their Grandparents the Houghtons.
I think you meant their grandparents, the Dawsons.
1871 census Wiliam and Thomas were in household of grandfather, James Dawson in Bedford Leigh, Lancs.
https://www.familysearch.org/ark:/61903/1:1:KZP8-5QXWilliam was in Pennington with his Leigh-born wife in 1881.
https://www.familysearch.org/ark:/61903/1:1:Q27T-NKX4There was a Mary Dawson, aged 44, widowed, a laundress, in West Derby on 1881 census. She was a visitor in the household of Francis Pearce. & wife. All born Lancs.
https://www.familysearch.org/ark:/61903/1:1:Q27L-GTKZMary may have remarried or lived with a man and took his name without marrying and Ellen may have taken the stepfather's surname.
As Mary and Ellen were "in and out" of the workhouse as jon discovered, the board of guardians may have arranged for Ellen to be trained or apprenticed somewhere. She might have "lived in" her employer's household or boarded out. There were industrial training schools. Since Mary was Catholic there's also a possibility that Ellen was put in a Catholic orphanage.
Diocese of Salford - Catholic Protection & Rescue Society was founded 1886. Website article "Former Children's Homes" has information about orphanages and industrial schools and a list of institutions.
www.formerchildrenshomes.org.uk/1911_institutions-_ _by_rc_diocese.html
Book "Leaving the Workhouse - The Story of Victorian Orphanages" by Gudrun Limbrick is featured in the article.
Peter Higginbottom's workhouse website has a section on Bolton workhouse
www.workhouses.org.uk/Bolton/ Hollins Cottage Homes for pauper children set up 1879. Ellen may not have been classed a child by then. Bolton workhouse was one of the pioneers in cottage homes for children. Census transcriptions are on the website. The master and matron had a 2 year-old adopted son on 1891 census. I wonder if he was an orphaned inmate they'd decided to keep.
Some surviving school registers from 1870s have been digitised and indexed. Admission rolls include address and name of parent. I don't know if there are any for Bolton schools.