Waking up after heart surgery is nothing like waking up after any other surgery: it takes a very long time. You'll mostly be floating in and out of consciousness before you fully wake up, probably for several hours. In addition to the anesthesia, you'll be sedated, so when you do finally wake up, it's not like the fully awake feeling like you have now.
Were it not for the taste of plastic in my mouth (and knowing beforehand), I probably would not have known I was still on the ventilator. It didn't feel bad or uncomfortable, and this was a concern for me, too, before surgery. After being awake for about an hour, the nurse removed the tube; it was 5 or 6 pm, quite a few hours after surgery was over. It didn't hurt or cause any discomfort. She also gave me a cup of ice chips that were very enjoyable.