A few years ago I was persuaded by my mother to jump off a pier into the sea. It was summer. My mum is a swimming teacher of almost 25 years at that point, and regularly enjoys wild swimming. She was in the water below, having already acclimated.
The Moment that I hit the water, my entire body went into panic mode. I couldn't sense direction, my lungs were fighting against the breath I was holding and my limbs began shaking so hard I couldn't make the correct motions to actually swim.
It didn't matter that the water was probably 15°c (59°F) and comparatively warm. The on-land temperature was probably 30°c (86°F) and this is the UK, so it always feels hotter than it is. My core body temperature sits at around 36.5°c (approx. 98°F).
Swimming pools, for those who don't know, are generally kept around 25-30°c (77-86°F). Lower for competitive swimmers who will generate lots of heat while they swim, and higher for children who cannot regulate their body temperature as well as an adult can.
My mother grabbed me and pulled me up to the surface, and continued holding onto me while treading water herself, until I was able to control my body enough to swim the maybe 15 metres (50ft) to get back to the pier to climb out. It took several minutes to control my breathing and limbs enough to do so.
If she had not been there, I would not have jumped. Because without question, I would have died.
I am not a weak swimmer. Cold water shock Does. Not. Care.
If you absolutely have to go in cold water, do it slowly, up to your ankles, then knees, then hips, then waist, then armpits, then neck. Do Not move to the next stage until you feel relatively comfortable with the stage you are at, at minimum you should be breathing relatively normally.
Cold water shock is very real and utterly terrifying. Cold water does not have to be that cold. Treat bodies of water with the fear and respect they demand, otherwise they will reclaim you.