I'm sure that the SAR experts on here can correct me, but my understanding is as follows:
The UK has agreed under international treaty (Chicago convention 1950s springs to mind) to provide SAR provision within a specific geographical region. This is the responsibility of the department of transport. The DofT then delegates this task internally, which is where the MOD comes into the picture. Post WW2 it was the military who had a robust SAR organization already in place, for the rescue of downed aircrew, and with helicopters in their infancy, only they were really capable of providing the coverage required to fulfill our international commitments. The rest is history....
Standing by to be corrected, informed, hopefully in a constructive way....