It's too soon to worry about. Currently you still need to keep your UI in JavaScript / HTML / CSS so Web Assembly isn't going to be replacing that to deliver "fat apps" just yet.

Regardless, when there does come a time that WASM has full DOM access you'll just be replacing a ton of JavaScript and other code with WASM bytecode. Today you can already transpile many languages to JavaScript and some frameworks even allow you to never even touch JavaScript and still create web applications. I don't see this being much different with the exception that you'll no longer be required to used or transpile to JavaScript and this could end up faster (that's a big COULD and will probably depending on what you're doing and how).