If you are asking why I think it may happen rather than why it would be a good idea: imagine that you are a compiler writer for language X. You can target WebAssembly and run the language not only on the web, but also cross platform on servers and desktops and potentially mobile, and you even get a cross platform standard library including networking, UI (DOM/HTML), OpenGL, and more. This will save a tremendous amount of work compared to targeting LLVM separately and implementing your own cross platform libraries (which requires expertise on each platform that the compiler writer is unlikely to have). It's like targeting the JVM but lower level (compiler writers like this) and runs seamlessly on the web like JS.
Browser WebAssembly implementations will compile really fast because it has to be loaded and compiled for use on websites, making it suitable for a really fast compiler for interactive development and REPL, while it will be possible to compile to fast code via LLVM.
why do you say "as long as you don't use C or C++" ? these two languages are especially good when it comes to compile a single code base to multiple platforms. Got a 150kloc C++ app that "just compiles" to windows, mac, linux, android, iOS, under x86, x86_64, ARM, and it even worked on PPC macs last time I tried.
Multi-platform makes sense client-side, where you have to deploy to Windows/Mac/Linux/FreeBSD/Android/iOS on x86 and ARM.
You know your server arch. You know your server OS. As long as you don't use C or C++ (or you're carful to be multiplatform), all switching is a recompile away.