Had to do some maintenance in my swimming pool recently; the water temp is about 7C and I simply couldn't believe how quickly my body went numb on entering the water.
Did it in the end by standing in a hot shower wearing a tracksuit, putting lightweight waterproofs on over that & then getting in the pool.
I can easily believe you wouldn't be able to function in any meaningful way once immersed in the North Sea.
I always carry a liferaft when crossing the channel in a single but tbh I'm not optimistic about my chances of successful deployment and entry should the worst happen.