If I recall correctly, UK SAR is partly done by the Military (RAF and RN), partly done by the Coastguard, and partly done by private contractors (companies like Bond Helicopters and Bristow Helicopters). I'd need to do a bit of investigation to find out when the private contractors came on the scene though...in a T2K World it might still have been 100% Military.
Personally, I would have thought after the nukes started flying any available helicopters would be transferred to more out and out military roles. I think the likliehood that any military personnel would continue to carry out SAR duties if ordered otherwise is slim...not only would they be going against their orders but they would, presumably, be reliant on higher headquarters for more fuel once their original supplies were exhausted.
There is a civilian organisation in the UK called the Royal National Lifeboat Institution, which consists of civilian volunteers who man lifeboats; they can be found in a multitude of small towns and villages all along the British coast, and it's highly likely that they would remain true to their ethos, again provided they could maintain a supply of fuel.
More details on the RNLI can be found here
http://www.rnli.org.uk/
One point you have made me think of Mo is the organisation Medecins Sans Frontieres (apologies if I have spelled that incorrectly!). I'm sure that these people, and others like them, could be found in conflict spots throughout the T2K World, and would make an excellent cover for the activities of French Intelligence...