That will never happen for Julia, because the language is designed to be fast for scientific computing... targeting web assembly is a pointless additional layer of abstraction that only negatively impacts performance.
I see value in Julia web notebooks, where the lifting is done by a server (or localhost), but what are the motivations for running Julia in a browser?
Interestingly, there's a slow-but-steady effort underway to implement a generic BLAS and LAPACK in Julia itself, since this allows for computations with custom datatypes that aren't supported by the Fortran libraries. For example, you can now do QR factorizations and solves with any numeric type, including Rationals and Quaterions: http://stackoverflow.com/questions/20985783/rational-matrix-...
Julia's dispatch system allows for these kinds of layered algorithms very nicely, and they could just as well live in separate external packages.