I'm dropping her and returning to the pre 1538s.
Do come back, Aceh. It's easier to make progress in the 19th century then that far back!
The birth was registered by Joseph Hale, father
So he obviously existed.
Funnily enough, since Susan (aka Sarah) says she was born in St Clement Danes, there is a Joseph Hale there in the 1841 census! Age 35, Shoemaker, born in the county.
Perhaps they weren't together when the census was taken.
Re a death.
We know the Hales were in the East London registration district.
If we take Frances Hale, mother Williams, 1843-1844, as one of theirs, then there is a change when William's birth was registered in 1846.
These may be the sons' birth registrations and my guess is that there is no father named on either.
HALE, WILLIAM -
GRO Reference: 1846 D Quarter in EAST LONDON UNION Volume 02 Page 231
HALE, THOMAS -
GRO Reference: 1851 J Quarter in EAST LONDON UNION Volume 02 Page 228
Mary naming her father as John instead of Joseph is probably because she had no memory of him
Of course the Williams maiden name is back later on the births of Henry and Edward, but that just means that mum gave fuller information about herself. There may be no father named on the certificates. Joseph being named on the baptisms of the boys may have been done for propriety's sake (of course she may have fibbed on those last two certificates as well!)
So if Joseph had died before 1846, rather than just disappeared, there is this possibility in the quarter following the one for Frances
HALE, JOSEPH
Age at Death (in years): 42
GRO Reference: 1844 S Quarter in EAST LONDON UNION Volume 02 Page 167