For autoplay to work on Mobile Safari with a native HTML video element you need to also set the “playsinline” attribute. Otherwise the video wouldn’t start to play automatically:

<video muted loop autoplay playsinline preload="auto">
    <source src="video.mp4" type="video/mp4">
    Your browser does not support playing this video.
</video>

The playsinline attribute is actually a standard HTML attribute, I learned today. I did not encounter any problems with autoplay in other browsers besides within Mobile Safari though.