After Hurricane Charley, as other people have pointed out, the MK was open the next morning at 9am, which I think was right on schedule. The only things different from a normal August day in the park were that the crowd was somewhat smaller, Fastpass wasn't working, and in the wooded area near Cinderalla's castle we saw a single tree stump where "elves" (or somebody) had cut off a fallen tree flush to the ground sometime between the worst of the storm at 9pm and park opening at 9am. There were no wood chips lying around the MK, no broken tree limbs, nothing at all out of the ordinary. Phenomenal!
Typhoon Lagoon was still closed on the second day after the storm, but Blizzard Beach was open. I assume that the water parks took more work to clean up because of the thousands of trees in them. At BB we did notice a fair number of tree limbs which had been cut off flush with the trunk, and a few tree trunks cut at the ground.
Outside of the theme parks, the WDW property (and the whole Orlando area) had many, many downed trees. They had been cleared from the roads but were lying in the ditches and roadsides.
Off WDW property there were a lot of big billboards blown out, and quite a few large and small buildings that had large chunks of siding or roofing blown off. This must be why the hotel staff didn't want anyone going outside during the storm - if you were out filming the horizontal rain and a large sheet of broken glass or metal roofing came flying along in the breeze, that could ruin your day!

During the height of the storm (we were staying near I-Drive) when the wind blew at about 100 mph for a few minutes, the air was filled with flashes like lightning or fireworks - this was the flashing of electrical sparks from billboard signs being blown out, and from overhead electrical wires touching and shorting out.
One thing I will never forget is that at the very height of the storm, with 100 mph wind, horizontal rain falling in massive buckets, trees falling over and glass and metal debris blowing around in the air like confetti, there were still people driving around in their cars, and running outside so they could take pictures of each other leaning into the wind. I guess it's true what they say, you can actually die of stupidity. The TV news people said they were stunned to see people speeding down I-4 in the middle of the hurricane and the next day they said that some of these people were killed when their car wiped out. I remember one reporter wondering out loud, where could these people being going?
To make a long story short, I would recommend not worrying about hurricanes too much because they don't affect Orlando very often. If you are unlucky and one hits, make sure you keep constantly updated so that you know when and if you should prepare to hunker down in your hotel with some bottled water, candles, snacks (just in case). And stay indoors and don't drive around in the middle of the hurricane fer cryin' out loud!