I think that this is one of those cases where you can either spout "air law" (which in this case has nothing to do with it) or consider the practicalities.
I agree completely which is why my next sentence was that personally I would call 121.5. Knowing that you used CAP413 correctly will be of bugger all use to you as you slip into a hyperthermic coma.
The point about time to get rescued is a very valid one. I believe the nearest helicopter base for SAR is now Wattisham. Quite incredible that the nearest base to the world's busiest shipping lane is an hour away.
The Lille option definitely seems the best and I will be using that from now on. Thanks 2D, that advice may just save someone's life
