You have iTunes set to delete the show(s) after some amount of time. This probalby needs to be done as they take up a lot of space and can quickly fill up your iPad or computer. When you bring up the show on Apple TV are you looking under "TV shows" or "Computer". If you want to view your local content you need to look under "Computer". Everything under "TV Shows" on AppleTV is streamed from iCloud. Last but not least even if you do have to stream it the show should start in less than a minute after it is selected unless there is a serisus network connection issue.