If your licence is valid in the UK and is the Pink Plastic with Photocard ( and you have the accompanying rap sheet), you in theory need to do nothing, however some Guardia officers will insist that you register the licence, although under EU directive as I understand it this is not necessary.(I think on one of the threads here there is a letter that explains the current EU legislation and its validity in Spanish, which is worth copying off and using if challenged
In the UK as long as you have the two part licence, the hire company will have little or no interest in what address it has,
I have been driving on my UK in spain now for Six years, use regularly in the UK for hire cars, ( on my old UK address) have been stopped and fined for speeding here in Spain and there has been no issue other than I had to pay the fine!!,