Looking at 1871 census, for William (& variants) Haywood (& variants) living in London, and born in London I can only find:
William Haywod, age 35 at St James, Westminster
Sarah Haywod, wife, age 33
together with 2 sons and 2 daughters.
(RG10/142/28/57)
Repeating the process with Hayward (& variants) comes up with:
William Hayward, age 36 in Hackney, a Clerk, b St Edmunds
Jane Wallice, wife, age 26
(RG10/324/18/33)
William Hayward, age 34, Visitor in Mile End Old Town, a Newsagent, b Spitfields
(RG10/564/45/35)
Kevin