As obvious as it may seem, the first thing to do is check that your settings look like this:
If your settings look good but you are still not seeing your video autoplaying on your mobile device, there could be a couple of reasons why:
(iOS) Your power saving mode is on. Enabling power saving mode prevents ANY video from auto playing, as your phone is trying to save as much battery as possible. Go to your phone settings, disable this option and test your video again.
Your data saver is on. Just like on the previous case, this will prevent videos from auto playing on Android.
If you have an animated GIF thumbnail plus either of the previous options enabled on your phone, your video won't auto play, and, in fact, it will show a black or white box.
To prevent things like these happening, we recommend always enabling the big play button on your videos; even if they auto play. This way, your website visitors will always know there's a video to watch, even if they are trying to save their battery life.
Note regarding iOS 9: videos won't autoplay on iOS 9, no matter what video provider you use. This was a feature of the software at the time, that Apple decided to change on later versions. You can read more about it here.
Note regarding data saver mode: Data saver mode has a known conflict with disabled pausing, causing your video not to play, so make sure you use disable pausing when you know it's necessary! As you can see in this case study, disabled pausing may even negatively affect your conversions, depending on what your video is about ;)