Almost
I am sorry this is becoming more confusing than it needs be.
If you file a FPL into the UK to an airfield that is on the AFTN or can receive your FPL details via another means, and most importantly, a DEP message, then they will initiate overdue action if you fail to arrive within 30 mins of ETA (and full overdue action after an hour). You don't have to 'nominate' them to do this. When you arrive at this destination you need do nothing about closing the FPL, as it is deemed 'closed on arrival'.
The problem only arises if your destination cannot receive your FPL details and be expecting you, you need to ask someone to be the Responsible Person. It is this person who needs to alert the authorities of your non-arrival and therefore needs to be told of your arrival if he or she isn't waiting for you at the airfield. If you have to ask the Parent ATC unit (or other ATC unit) to be that 'Responsible Person' then you must inform them withing 30 mins of your arrival otherwise they will commence overdue action.
Any clearer?
As an aside, I have been asked to close a FPL by someone who we were acting as the Responsible Person over the RT ("I'm about to land, can you close my FPL?") Just in case he crashed and burnt I asked him to ring us on landing.