That will take a while to come together. Key blockers are removal of the feature flags currently needed to enable things like garbage collection in Firefox and Chrome and the completion of Kotlin 2.0 and it's new compiler (K2) which is currently available via a feature flag in the Kotlin 1.9 stable release. My estimation is that we're about 6 months away from those blockers being resolved. IOS support is likely to transition to beta around the same time. From there to being stable and well supported is probably another year or so.
But a lot of stuff works right now already. However, it's not suitable for production usage because of the early development status the various bits and pieces you need and the need for toggling browser feature flags.
There are some nice examples of IOS support in the recent release notes for compose web: https://blog.jetbrains.com/kotlin/2023/08/compose-multiplatf...
They don't really mention Wasm there as this mostly focuses on the IOS support. They had a lot of presentations about that at kotlin conf and the compose web channel in the kotlin slack is very active.
Basically, anyone currently doing mobile development that is used to modern UI frameworks for that, will soon be able to target browsers effortlessly without compromising on their UI frameworks. Compose is one of the frameworks. But there are others. I've seen some nice kotlin-js frameworks targeting canvas and vector graphics. Doodle is a nice example: https://nacular.github.io/doodle/. I have not used that yet but it looks quite slick. A lot of kotlin-js stuff will transition to wasm once the compiler stabilizes.
Web developers seem to be mostly unable to see beyond their comfort zone of DOM/CSS/JS. There are alternative ways of doing UI/UX that are common outside of browsers. Applying that in a browser is transitioning from impossible (a few years ago) to being hard but very feasible (the last few years) to being easy, very common, and widely supported across different developer ecosystems (the next few years). Not a matter of if but when.