1662 is just 2 years after
restoration of Charles 2nd to the Monarchy and the hanging of captured
regicides, Some STAPLEYs were involved with both sides of the civil
war, Many records
were lost or destroyed during that war.
In 1665 there was an
outbreak of Bubonic Plague [The Great Plague killed an estimated 100,000 people, 20% of London's population] followed in September 1666 by the Great Fire
of London which destroyed much of the centre of London.
William had a sister: Mary baptised at Ticehurst 26 july 1659, so it seems unlikely that Anne Blowers was William 1662's mother.
I think it more likely that the father was William of Horsmonden (4th cousin twice removed of the regicide) orphaned at age 11,
or William of Bodiam orphaned at age 15. Both of them would have struggled in the 1660s.
See the photo gallery of his descendants