As of a few days ago, IOS support in alpha and will hit beta next year. Android and desktop support are now stable. Once that stuff stabilizes, you can write UI applications that basically work on any platform.
The wasm compiler will roll out along with kotlin 2.0 which is the next major release for Kotlin and features a new compiler (k2). This looks like it should happen early next year. k2 is currently available in beta release and you can enable it in kotlin 1.9.x.
The nice thing with the Kotlin multiplatform ecosystem is that there are already a lot of libraries that work across the various platforms. So, the wasm compiler will rapidly become part of that and inherit a lot of nice libraries. Mostly all that takes is for builds to be reconfigured to target that ecosystem and implementing some platform specific behavior where that is missing.
Another interesting thing in this space is using and linking libraries written in different libraries. For example a lot of the platform specifics are probably going to be dependent on e.g. C or Rust libraries that are already available. In many cases these might be the same libraries that Kotlin native uses.