I think the question is just too subjective to answer. In the end, there are excellent C++ libraries for nearly everything that has to do with the solution of PDEs, whereas they are largely missing in the Julia environment.
Examples that come to mind are PETSc/Trilinos for linear algebra, deal.II/libmesh/FEniCS for discretizations, etc. You will have to duplicate many many many years of work if you want to do all of that in Julia. Is it possible? Yes, of course. Is it worth it -- no, not at all.
To give you an idea of the level of work necessary: My best guess is that every finishing graduate student's software project based on deal.II runs through maybe 200,000 lines of C++ code in deal.II and, if it uses any kind of interesting linear solver, another 100,000 lines of code in solver packages. But experienced, full-time programmers only write 20,000 lines of code per year -- in other words, these students' programs would have taken 15 years to write if not for existing software libraries.