Rubber dinghys (life rafts) for fighter pilors didn't become standard issue until, I think, late 1941, and even then, certain types of aircraft at first could not accommodate the packs, which were fitted to the seat-type parachute cushion. In the Mustang III, (P51B/C, 1944 on), the pilot wore a back-pack parachute, possibly of American manufacture, as the G.Q. and Irvin British packs were seat type. This meant that the 'K' type dinghy could be fitted to the seat strap 'd' rings of the parachute harness. but, until then, most fighter pilots had to rely on a rather antiquated 'Mae West' life jacket, early models of which had to be inflated orally. The slim, lightweight, CO2 - inflated Luftwaffe life jacket was much prized by R.A.F. aircrew, one of the most famous to use one being Guy Gibson.
As for the seas around Britain, you are quite right, the North Sea is colder, even though, in geographic terms, it is classed as 'warm', due to the Gulf Stream; however, this is more relevant to surface temperature. Even a few inches below the surface, the temperature is considerably colder and, after sudden immersion, with heavy clothing saturated, the effects of the penetrating cold cause rapid cramps, leading to muscle spasm and rapid exhaustion, which accelerates the cooling of the body's core temperature. The English Channel is not much better, and is renowned as a cruel and treacherous stretch of water; ironically, allied aircrew probably had a better chance of survival if downed nearer the enemy coast, as the Germans had a chain of 'Rescue Buoys' anchord offshore. These were large buoys, with a superstructure resembling a small submarine, on board which was heating, food and water, a comms radio, and seating and bunks. These were patrolled by the Seenotdienst, the equivalent of the Air Sea Rescue service (as it was known up until the late '50's / early '60's.) by launch and seaplane.
As the war progressed, obviously the 'survival' equipment, and training, improved, and fewer lives were lost. But, even in the 1960's, with the help of (then) modern helicopters, Air Sea Rescue could still be a delicate affair. There is a story (true) of how a downed pilot was only spotted, at the last minute, by a launch crew member, because he saw the reflection of the pilot's wrist watch in the water, just a few feet away!
Finally, just to illustrate the effects of the British coastal waters, here's two stories. Some years ago, a friend of mine was flying from Barton, near Manchester, UK, to I believe Belfast, in a Bell Jet Ranger helicopter. Half way across the Irish Sea, he experienced a severe malfunction, and had to autorotate to a forced ditching in the sea. He was able to transmit a 'Pan' call, and survived the ditching, ending up floating in his (modern) life jacket, and clad in the immersion suit he always wore on this regular flight. However, the cold water had such a rapid debilitating effect, that he was unable to climb into the life raft, which had inflated alongside him. Very fortunately for him, his distress call had been received, and a R.A.F. Sea King SAR chopper arrived within about ten minutes, just in time to grab him from the sea, semi-conscious, and already in the early stages of hypothermia. He spent a week in hospital recovering.
The second story involves an area of sea not that far distant, around Hollyhead Bay, on the west coast of the isle of Anglesey. Here, trainee aircrew from R.A.F. Valley, are 'dumped' in the sea, as part of their training, to be 'rescued' in a combined exercise with the RNLI lifeboat, and the SAR Sea Kings, which replaced the Wessex choppers in the early '90's. These aircrew are 'dragged', in full flying gear and parachute harness, to simulate a parachute landing in water, with a surface wind dragging them across the waves. After releasing themselves from the harness, they are left, sometimes for 15 minutes or more, before being recovered. This is to simulate, and acclimatize, or mentally prepare, the students, in case they ever have to do it 'live'. Ask any of these poor sods what it's like, and they will tell you they thought they were going to perish, due to the cold, strength-sapping water.
I did a 'water jump' once, and never again, into the sea around Liverpool bay, in summer. The sudden shock of the cold water almost prevented me from getting out of the parachute harness and, bear in mind, the drills during the descent ensure that you are virtually free of the harness even before splash-down!
Sorry if this has been a bit long-winded, but I hope it's given at least a little insight and been helpful.
Terry.