My theory is that Sarah was unmarried, she said she was a widow for appearance's sake. Either the children were unregistered (except possibly James), registered under father's surname (which may not be the same for all children) or given the itinerant lifestyle not registered in the district where they were born.
Sarah said she was born in Cambridge, Cambridgeshire, again she may not know with any accuracy where she was born, if her mother / parents were also moving around a lot. But it could well place her in Cambs rather than Essex.
My best candidate for Sarah is one born in Royston, living in Barkway Road Royston in 1871. At this time the Herts / Cambs boundary chopped the town in two. It was moved north (controversially) to give the northern part to Herts about 10 years later.
Sarah age 4 is with her mother (also Sarah age 32) and siblings Mary Ann age 8 and John age 1 (it looks as if 15 months has been scratched out). They are living with Edward Forby (?) age 40 described as lodger, unmarried, occupation hawker. Sarah is also described as lodger, her status is described as mother which has been crossed out an ditto added, i.e unmarried same as Edward above. There is no head of household, Edward is the first named as occupier, but it may have been so temporary occupation he described himself as lodger. The house is the third one in the Barkway road list after the end of the High Street. I suspect they've been demolished for the one way system and the "bus station". There are some small terraced houses prob mid Victorian a little further up Barkway road, which could be of similar type.
Sarah senior is said to be born in Melbourn, Cambs which at the time would have been on the direct road from Royston to Cambridge. Mary Ann was also born in Melbourn, Sarah jnr and John in Royston.
Given age, location and occupation of mother's partner (and possibly natural father) this could be the Sarah who was in Tilbury in 1891.
added
I think Edward Fourby is actually Edmund Yourby, I think I have found him in 1881 living in Cambridge (St Andrew the Less) with wife Mary Ann 44 b Steeple Morden Cambs and daughter Elizabeth 10 b Royston. He is still born Potton Beds and still a hawker.
No sign of Sarah Livermore and children, she could have moved in with someone else and either married or adopted his surname.
In 1861 Edmund Yourby is living with Sarah Livermore (no children yet) in Melbourn. He's a gardener and she is his "lodger"
