The UK's already got SAR helos, but due to the size of the country, a force on the sheer scale of the USCG would be overkill.
Ha - you've been reading what I wrote on the British technology thread, havn't you?
It's difficult to get Britain to have a coastguard - the strength and availability of the Royal Navy has always precluded it.
Nevertheless, maybe you could work out a POD around something like the Cod wars. Perhaps a frakar between a Royal Navy ship and an Icelandic coast guard vessel goes wrong. The RN ship accidentally rams the icelandic ship too hard causing it to sink, taking several crew members with it. Several days later photographs of the incident showing a large British warship sinking a small civilian fishers protection vessel cause outrage. In response, the government decide to hand over fisheries protection to an expanded coast guard. Over time the coast guard expands to include ocean tuboats for emergency tow and ships for customs and border patrol. SAR heclicopters might also be a possability.
Russell
Just because there isn't an independent entity like the US has doesn't mean the same jobs aren't being done, it's just that other groups are doing them.
If all you want is the name coast guard attached to something flying a Union Jack, as Tovarich pointed out that already exists.