When it come to legally changing your middle name the process is quite simple:
Remember, until you legally change your documents, your new name will not be recognized as your legal name. Mere possession of your deed poll does not give your name a legal recognition.
Of course, there is no requirement that you immediately change all your legal documents like drivers license, passport etc. However, if you are planning on migrating to another country, it is imperative for all your documents to have the same name on it.