I would suggest that you go to the documentation office of the National Police in the area where you live (probably that was where you got your NIE) and tell them you are now married and wish to change the name. When you have that then I would assume you then need to go Traffico with ypu new NIE/EU citizens registration cert and get your DL changed too.
Of course in Spain, Spanish nationals do not (cannot) change their name on marriage. It is just a custom in UK , as there is no law which requires any change of name.
My wife decided to change her name when we married, she is Non EU. First, she changed her name of her passport and then when she renewed her Residencia, the passport name was put on the Residencia. NO problems arose