It looks like Thomas senior did remarry.
In 1881 he's at 19 Mayo St Shelton, described as stepfather of Richard Lilley.
There's a marriage in Q2 of 1872 mentioning Thomas Allcock and Mary Lilley.
The image is on FindMyPast - St. Peter ad Vincula, Stoke-on-Trent:
May 13 1872
Thomas Allcock, 45, Widower, Gilder, Peel Street, Thomas Allcock, Turner
Mary Lilley, 45, Widow, , Charles St Hanley, John Phillips, Fitter
(both signed)
Witness: George Sims, Mary Ann Simms
Of course all this hinges on whether the Preston marriage gives a matching father!