Background videos do not autoplay on mobile devices due to mobile device constraints.
However, one of our community members found a custom code work around here: HTML5 Video autoplay now working on mobile? - #2 by vincent
If you have any trouble with the above integration, can you please make a post on the Webflow Help pages under the Code Help category with as much information as possible (read-only link, screen shots and a description).
Another option is to add a background image that will display on mobile instead of the bg video.
Hope this helps! 