On FindMyPast.com, I did not find a matching christening record for a William Drake, son of William, who was born in 1553 in Essex. There was a William Drake, son of Thomas, who was christened 27 Jul 1552 in Little Waltham, Essex.
I did find the 14 Sep 1555 burial record for his alleged father, William, in White Notley. I also found other misc. christening records from 1552 in White Notley.