I have never attempted suicide, but this would be my preferred method if I do. However, I can tell you about my experience of mild hypothermia from cold water while attempting to cross a river in Iceland. The water was fast flowing and its temperature was close to freezing. Air temperate was not warm, but was above freezing. I was knocked off my feet by the current several times, and became thoroughly soaked. I was on a rope, for safety, and my companions eventually dragged me back to shore. By then I was barely able to help myself. You lose your strength very quickly in that situation (so you might not be able to get out of the river if you wait too long). It was not painful, and no more unpleasant than cold water normally is. 2 degrees C would probably be cold enough to die, if you were soaked through, but it would help if there was a wind, or at least some breeze.
However, if you do attempt this method, and especially if you are in good physical condition, you might wish to consider going without food for several days (up to a week) beforehand. That would deplete your body's energy reserves, making a comeback less likely.
A good thing about the sorts of temperatures you suggest (slightly above freezing point) is that if you fail you won't have to deal with the horrors of frostbite.
