Hi there - and a very warm welcome to Rootschat!
![Cheesy :D](https://www.rootschat.com/forum/Smileys/classic/cheesy.gif)
Just for your records, here are the occupants of 33 Alpha Square in 1881...
1881 Newington (St Peter Walworth) London
RG 11/540 folio 42 page 1
Frederick J Cole, head, 51, cabman, Yeovil Somerset
Sarah, wife, 47, Middlesex London
Ellen, daur, 23, book folder, Middlesex London
Frederick W., son, 17, brass founder, Middlesex London
William, son, 15, Middlesex London
Jesse E., daur, 19, Middlesex London
Amy S., daur, 10, Middlesex London
As regards the whereabouts of William and his mother Maria, I suppose it's not unlikely that she married shortly after his birth and that they are listed somewhere under her married name.
Checking the parish records is a good idea, although I think the London Metropolitan Archives are closed for stock-taking/refurbishment until January - you may have to wait for a response.
Regarding the 1911 Census, no you can't view it at The National Archives. At the moment you can only order a copy of the household schedule for a specific address (£45).
And I yes, I would certainly obtain a copy of William's marriage certificate. If he knew who is father was, he may have named him. If not, he may have invented a name or left it blank.
Dave