It looks as if Sidney's only brother who survived to adulthood was also called Robert (Robert Dilley Green) and also married a lady with surname Green. It's looking more as if Sidney lived with Sarah Ellen and fathered James, but then left her to marry Elsie in 1935.
Still no sign of Sidney in 1921.
I think this seems very feasible. Sarah Ellen moves location within St Pancras/Islington after 1934. it may be coincidence that she moved just before Sid married another woman in 1935 but I think it's doubtful.
Sid then changes his name on the 1936 electoral register to WILLIAM Dilley. Elsie is with him. 1937 sees him reverting back to Sidney.
Sid's DOB is out by 11 years on his death certificate. Elsie obviously knew very little about him.