Isn't it just a real GNU/Linux system on ARM? So couldn't you just go grab the official syncthing binary and runn it yourself from a terminal and use the web interface?
I suppose that's an empirical question. I'm not sure what happens to apps run from the terminal in Ubuntu Touch when they're not the currently active application (i.e. whether it will run properly in the background). Having an app-ified version would be ideal though, so I could do things like on Android when I have syncthing disabled unless the device in on AC power.