Use SymPy.jl. SymPy can parse Mathematica expressions, and SymPy.jl can convert SymPy expressions to Julia expressions. For example, if we define:
using SymPy
const sympy_parsing_mathematica = SymPy.PyCall.pyimport("sympy.parsing.mathematica")
mathematica2julia(s::AbstractString, substitutions::Pair{<:AbstractString,<:AbstractString}...) =
SymPy.lambdify_expr(sympy_parsing_mathematica["mathematica"](s, Dict(substitutions...))).args[2]
then we get:
julia> mathematica2julia("Log[Log[x]]")
:(log(log(x)))
julia> mathematica2julia("Exp[Log3[x]]", "Log3[x]"=>"log(x,3)")
:(x ^ (log(3) ^ -1))