I'm not sure if there's a .NET-compatible implementation or interop but it'd likely take over C# as well if so, if Microsoft decides to invest in it. They have pretty good languages like F# already but Rust is designed around zero-cost abstractions and avoids the garbage collection penalty that comes with running on the JVM or .NET.
Rust offers performance, type safety, memory safety, and metaprogramming that are provided by few other (largely academic) languages, while permitting multiple programming styles and escapes from those restrictions when you need them.
On the other hand, I don't think the Web-oriented languages will go away unless someone develops a variant of Rust or Rust framework intended for building websites. However, for everything else that's not running inside a web page or supplying the content in a web server, I think we'll see a gradual transition to Rust; the web server will be in Rust; the JavaScript compiler will be in Rust; the browser will eventually be in Rust; and so on.