It's hard to give you a definite amount of time that it would take for the hypothermia to set in without knowing the temperature of the water that the dog was in. A dog becomes hypothermic in water much faster than they do in air, which is why the temp of the water is important.
The normal body temp for a dog is 101-102.5. Once the body temp reaches 99 degrees, it's the onset of hypothermia. In an 11 pound Yorkie, that could kick in within 5 minutes if the water was cold enough.
Being 'numb' means that the nerves that register pain get, for lack of a better word, shut off, meaning that the dog wouldn't feel the pain of the cold water that he's submerged in. As the extremities begin to freeze, it would be extremely painful, which is why the numbness helps.
As for drowning being painful or scary...I assume being unable to draw a full breath would cause a bit of panic, however, if your dog lost consciousness when that happened, he wouldn't have felt anything at all.