The problem, apparently, is that UIWebView does not include the same user agent string as mobile safari. (See here.)
If your video is converted to HTML5 or MP4, as follows from your comment above, you will need to change the User Agent of your UIWebView so that it matches Mobile Safari, so that your site knows to convert the view from flash (flash will not play on iOS). Here is the answer on how to change User Agent to UIWebView.
Notice, I have not tried this approach before.
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