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There is this 1861 census entry for a William Edward GOUGH bn 1859 grandson
but his birth is dittoed as Birmingham Worcester - I can’t see a connection to the surname Newton 

Piece:   2088
Folio:   42
Page Number:   22

Edward William   58
Hannah William   58
Eliza William   26
Charlotte William 23
David Williams   17
William Edward GOUGH 2 grandson
Charles Turner   17

Edit to add they had a daughter named Elizabeth perhaps she married a Gough 

Dec 1854   
Badham    Ann        W Bromwich    6b   1042    
Gough    Thomas Walter        West Bromwich    6b   1042    
Turner    Joseph        West Bromwich    6b   1042    
Williams    Elizabeth        West Bromwich    6b   1042

Thank you, it's strange where he has been from his birth to arriving in Ilkeston in 1881, possibly 1871 in sandiacre

Possibly of interest.  James and Rachael Gough have two daughters born after civil registration began, both are with them on the 1851 census. One (I'm unsure of her name; it looks like Rachael, but also looks different from her mother's name) aged 11 and Eliza aged 9. Both born in Ealing, which is in Brentford district.

GRO index shows a Rachel Gough Mmn Newton registered Dec Q 1837 in Brentford district, and a death of a Rachel Gough age 0 in March Q 1838.   In September Q there is an unnamed female Gough child Mmn Newton registered.  Eliza remains elusive, unfortunately - but at this early time not all births were registered.

James and Rachael Gough name their birthplaces as Heath, Oxfordshire, and Hughendon, Buckinghamshire respectively. 
A Rachel Newton daughter of Thomas and Rachel was baptised at Hughendon  on 17 April 1801 (Rachael is aged 50 in 1851).

I can't see a marriage, but if it was in Buckinghamshire, it may not be online.

Thank you for this, it all seems to fit, and makes a lot of sense

I see on that Ancestry tree that they have Sarah as dying in 1862,  the burial for that death is on Deceased online

SARAH GOUGH
burial date 19 April 1862
location Kensington And Chelsea
There is one other burial in that grave.

Thank you, I don't think that is her, she was still a live in 1871 in sandiacre, supposedly the wife of Robert Francis Thurston. With help from here it has also all fallen into place, she actually used her mums maiden name of Newton as she wasn't married.

Possibly of interest.  James and Rachael Gough have two daughters born after civil registration began, both are with them on the 1851 census. One (I'm unsure of her name; it looks like Rachael, but also looks different from her mother's name) aged 11 and Eliza aged 9. Both born in Ealing, which is in Brentford district.

GRO index shows a Rachel Gough Mmn Newton registered Dec Q 1837 in Brentford district, and a death of a Rachel Gough age 0 in March Q 1838.   In September Q there is an unnamed female Gough child Mmn Newton registered.  Eliza remains elusive, unfortunately - but at this early time not all births were registered.

James and Rachael Gough name their birthplaces as Heath, Oxfordshire, and Hughendon, Buckinghamshire respectively. 
A Rachel Newton daughter of Thomas and Rachel was baptised at Hughendon  on 17 April 1801 (Rachael is aged 50 in 1851).

I can't see a marriage, but if it was in Buckinghamshire, it may not be online.

I didn't know that, certainly something I can follow up, it's quite possible she didn't give the correct address of birth as it was in St James street kensington, and I was told it was a posh area

Have you found her or William on the 1881 census?

Not his mum, but he is now married to Rachel formerly Bostock and now living in Ilkeston which is just up the rd from Sandiacre where a similar William born In Nottinhill with his mum Sarah from Ealing registered as the wife of Francis Robert Thurston, for whom no marriage cert exists and Francis was living with his mum in 186o in Suffolk. All baffling