Miss Wood, 36, a mother-of-one, had met Appleton on Facebook, unaware of his horrific history of violence against women, including repeated harassment, threats and the kidnapping at knifepoint of one of his ex-girlfriends.
At the inquest into Miss Wood's death last year, Coroner Jennifer Leeming said women in abusive relationships should have the right to know about the violent past of the men they were with.
But the domestic violence charity Refuge has opposed the scheme.
Sandra Horley, chief executive, said: "It is highly unlikely that she (Miss Wood) was killed because the police didn't inform her about her ex-partner's violent history. It is more likely that she was killed because the police did not respond to her emergency 999 call for help.
"We are at a loss to understand why the Government is spending precious time and money – especially at a time of austerity – on this new scheme.
"As the law stands, the public already have the right to ask and the police have the powers to disclose information about a man's previous history."
''Sarah's Law'' named after Sarah Payne who was murdered by paedophile Roy Whiting in 2000, now gives parents the right to know of any child sex convictions of men with access to their children.
Mr Brown, a former prison officer originally from Aberdeen who now lives in West Yorkshire, said last month the ''world is watching for a lead from the UK's Government''.
A Home Office spokesman said: ''Domestic violence is a particularly dreadful form of abuse and we are constantly looking at ways to strengthen protection for victims.
''That is why we consulted on introducing a domestic violence disclosure scheme, often known as 'Clare's law'. We will be making a formal announcement shortly.''
Mrs May last year agreed to open a 'Domestic Violence Disclosure Scheme' to public consultation and is now considering the response.