I'd suggest you to check drejkim/edi-cam · GitHub. I believe that is exactly what you are looking for. However, I'm not sure if this method is capable of streaming 1080p video. The creator does use a HD camera (capable of 720p), so I believe you'd have to try it yourself. Just a couple of heads up:
1. Make sure your camera is UVC compatible. You can check it in Linux UVC driver & tools.
2. You will need an external power supply (7v-15v) for the Edison to function as host.
Since you are using Ubilinux, I can't assure you that it'll work since the guide was tested on Yocto. But, follow the guide and let us know what happens.