I quote from the National Archives at www.nationalarchives.gov.uk:
Despite what people generally think, it has always been possible to change your name without having to register the change with any official body. It is still perfectly legal for anyone over the age of 16 to start using a new name at any time, as long as they are not doing so for a fraudulent or illegal reason.Some of these changes were made or had to be made more official (e.g. if two parties were involved in it). In the eighteenth and nineteenth centuries Royal licences to a change of name were particularly common. Many of these licences were published in The London Gazette.
There I found the licence granted to Alexander John Sharpe to call himself and be called Alexander John Ellis. It's in Gazette number 18197 of the 24th of November, 1825. Here is the extract:
A change of your name may, at times, put you in the position of living unhampered by pecuniary sorrows. Mmh ... I guess I've got the wrong name. Any generous offers out there?