TWZ,
I think you'll find the Department of Transport, or whatever its predecessor was called, has been responsible for SAR within the UK FIR since 1944:
Convention on International Civil Aviation - Wikipedia, the free encyclopedia
See Annex 13.
Within the UK, as indeed in many countries, this international obligation has been met on behalf of the DofT by military assets. Initially only the military were capable, and there was no great desire to spend any money duplicating what the military could achieve.
Is the military/MOD today being honest to the DofT about what it can actually achieve now the Nimrod is no longer available. If I were the Minister of Transport I would be asking some probing questions, rather than believe all the mouth music being spouted by MOD about coverage provided by C130, etc...