There seem to be at least four things going on here, so simple answers are going to be difficult.
One variable is how you're building your article -- seems like you're using InDesign rather than HTML.
Within InDesign there are two ways to approach this: a video overlay or a web overlay -- each has different features.
An InDesign video overlay has the most control over playback (full screen option) and you can embed or stream the video resource. This is the option Benoit is covering above.
Once you go to code-based videos, more variables enter the experience. Streaming from a third party (YouTube/Vimeo) adds more variation as well. And then the interpretation of the HTML may be different for iOS versus Android. -- I suspect these last two aspects are what you're experiencing.
The "install Flash" problem is likely a poorly coded resource that's calling a Flash version of a resource rather than one encoded more neutrally.
I think the easiest route to a more consistent experience is if you used a video overlay, but that won't support a YouTube stream. OTOH you could embed or stream from your own https location if you have the video file.
Code based is going to have variable and somewhat more limited options. Variation may come from different OS support, and you won't have options like autoplay for the video.
HTH