Apple still control what API you can use in their mobile browser, and it's not like they are at the forefront when it comes to implementing Web API today, especially when they also forbid any alternative browser on IOS. Ultimately WebAssembly is limited by the API it has access to in the context of the browser (and the resources that are granted by the browser). All the promises of the "web as an app platform" have not been fulfilled yet.
IMHO WebAssembly will play a bigger role on the server, with nodejs for instance as it will allow portable native code distribution.
I'm not suggesting it's a way to build mobile applications without paying Apple. I'm suggesting it will be possible to deliver a product to (nearly) all platforms using 1) a single codebase and 2) bypassing walled gardens.
Personally I won't build products for an App Store anymore. They're too restrictive, expensive and time consuming to be worth it. But being able to, say, deliver a desktop product (either via the web or packed in something like Electron) and also deliver nearly the same experience to an iPad Pro would be huge.
Meanwhile I will keep my sandboxes.
"Write-once, run everywhere without paying someone for the privilege"
I think the term's overused, but this could actually be a "game changer".