We almost always go in Sept. A wonderful time of year to go, crowds-wise. Kids in the US head back to school in mid-late August for the most part. We find that the lines are super short (though we are planners and love Tour Guide Mike), and we generally don't wait more than 10 minutes for any ride.
We are from the US and are very used to hot and humid weather, so the weather in FL for Sept isn't much of a shock. Sept isn't nasty like July and August, but it can be quite warm and humid. We've had hit and miss when it comes to rain. Some years we saw almost none, some years there was still a shower around each afternoon.
Do remember that it is Atlantic hurricane season. Generally you should be fine, but keep it in the back of your mind.
Have a wonderful trip!!!