If someone were to actually cut their foot off like in the movie SAW what would start to happen following the cut and what would the chances of survival be?

I would say that an extreme infection would follow, then ganegreen would happen. Eventually that would spread throughout the body and you would die. I would advice against it!

Another answer:

First and foremost, Dont do it! But this is fiction. As a former paramedic, He would have to lacerate(cut) the skin and muscle and cut through the bone, which would take some time. He would sever the artery that feeds the foot and would have to stop the blood flow or bleed to death.

If he could withstand the pain and not go into shock, he could put a tournequet around the leg and stem the blood flow. If not he could bleed to death.

Since the cut would be with a saw, and not a scapel, he would sever many nerves and i would think he could not finish the cut, due to the pain factor, but if successful, he would have a 90-95 % chance of survival.

ok, now, what did he do with his foot? Take it with him?

He definitely would need fluids to replace blood loss, if not, he would get lethargic or confused and not be in his right mind, and could still bleed to death.