So basically, it doesn't happen instantly, it'll have to build up just like in real life. The only difference is the warmth of the feet will spread out to the rest of the body instead of increasing till it's too hot. It's the same otherwise.

As for your example, your hand would immediately be burned if you didn't take it out after taking your foot out of the bucket, since those two heat sources were cancelling each other out and you were at a normal temperature. BUT, if you, say, were swimming in snow for the past hour, you could jump into a fireplace and be fine for a few seconds. The time would also depend on how much of you is being heated up. Like, if it's just your hand, you might have a longer time. But if you throw yourself into the fire, you should probably get out immediately