A few little discrepancies in the facts at the moment,
i have found your family in 1901
William b 1863 birmingham
Sarah Ann b 1863 Essex
Ernest w 1892 America
William S 1898 London
the William you say is an Iron moulder in 1881 was born in 1853
There is a William born 1863
on the 1881 census living with his family in stoney Lane, birmingham
his father is Moses and both William and his father are Bedstead fitters.
The william in hospital in Birmingham in 1911 is also b 1853.
When Ernest marries in 1920 he states father William Day, deceased Bedstead maker.