I think typical ASP.NET applications likely won't use AOT in the near future. There is quite some "magic" in ASP.NET that I suspect won't work out of the box with AOT because it is heavily reflection-based. And while there are solutions for cases like JSON serialization, as far as I understand they require writing different code right now, so it's not automatic.
AOT is much, much more interesting for cases where startup time matters, and ASP.NET usually isn't such a case. But I can imagine that it will be much more useful and easier to implement for smaller, focused tools that don't use that much reflection-based framework code. CLI tools might be the ideal case for trying this out.