Some sources that might be useful:
en.wikipedia.org/wiki/Name_change#United_Kingdom
and:
www.thompsons.law.co.uk/ltext/change-of-name.htm
Basically, under English common law your name is whatever you call yourself/whatever you're usually known as, providing there is no intent to defraud.
But it's pretty difficult getting this point across, e.g. to schools, GPs, who don't know that. So the best thing is to use a statutory declaration or deed poll, but that will cost. And if you read the wikipedia entry (which seems correct to me), even a deed poll can't be retroactive in that it won't change any official documents already issued.
An anecdote regarding the common law option: Some years ago I worked for a Diocesan Registrar (the legal adviser to a Church of England bishop). I first learned of common law names when a query came in from a vicar regarding a bride-to-be who had been known for many years as Miss A, where A was the name of her stepfather. But her birth certificate had the surname of X, her biological father who she hadn't seen for years. The legal advice from the Registrar was that the banns, marriage register and certificate should show her as Miss A, because that was the name she was known by, but that the vicar was at liberty to add a note to the register of "formerly known as X". There were subsequently similar queries that came in from time to time and, personally, I was very happy to find out that in those circumstances people could be married in a CofE church using the names they wanted, that reflected who they were within their family.