WebM would not help you at all if you only want one codec, unless youre comfortable losing every iDevice immediately. If you want one codec, it's H.264. Native support on iOS and Android devices, and you can support Chrome via Flash (which is seriously making me think about switching back to FireFox).
Ok, now suppose that you'd like to spend developer resources customizing your video player. If you do this in Flash you support the entire modern PC universe, and can fall back to HTML5 for iPad kind of as an afterthought ("mobile version").
If you spend your resources trying to customize an HTML5 player, you leave a large percentage of PC users out, either those using IE, Safari, and users without cutting edge browser versions (if you want to be a pioneer of WebM), or Chrome and Firefox (if you use H.264).
It's a solved problem only when everyone supports one high quality codec, and with Apple and Google both major players in the browser space this will never happen. So Adobe and Flash occupy a sort of stable island in the middle of a massive tug of war.
So encode it to H264. It'll work in most browsers natively, and Flash reads it fine as a fallback.