Hi Russell:)
You are a patient one

I think I have found them 1851 to start with and there are other children as well. Dates and POB are a little off but other things match;
1851 Census
HO107/1470/103/25
2 Simpson Cottage, St Pauls, Hammersmith
William ANDREWS, 31,head,mar,corn chandlers servant, born Lambourne, Essex
Rosetta, 39, wife, mar,laundress, Shoreditch,Middlesex
Wm, 9, son, unm, Lambourne
James, 7, son, unm, Lambourne
Thos,5, son, unm, Lambourne
Rosetta, 1, daugh, unm, Hammersmith, Midsex
Eliza HALKNER, 22, lodger
Caroline Halkner, 19, lodger
Ann ANDREWS, 29, visitor, born Lambourne, Essex
Marriage;
Rosette THOMPSON
William ANDREWS
March 1841
Stepney 2 236
Purchasing marriage cert will provide parents names

1871 census as you say has Alfred 26, born Alridge Essex and John.G. 12, born Hammersmith, with them.
Alfred could be reg. ? Alfred Andrews possibly?
I have just found your TREE and see you are descended from John. G. 1859.
Does his marriage cert state parents William and Rosetta?
I cant find them anywhere to confirm any other details so decided to run with Ann ANDREWS, born 1822 as she is reasonably easy to find and is a possible sister of your William.
1861 Census, Ann is still in London and a servant, aged 38, born ABRIDGE LAMBURN, ESSEX
RG9/28/96/25
1841 Census
Lambourne, Essex
Address: Abridge
HO/107/336/6/Lambourne
Edmund ANDREWS, 55, butcher
ANN Andrews, 20
Adrain Andrews, 15
Catherine Andrews, 14
trish