Speaking as a compiler developer: compilers for standard languages make no attempts to guarantee constant-time properties, nor to provide any primitives that can be used implement constant-time guarantees. Indeed, the developers are likely to be moderately hostile to proposals to add such primitives--it is a pretty different beast if you're worrying about constant-time.
As such, if you're using a standard compiler for a language to implement a constant-time guarantee, you need to be doing verification of the resulting assembly to make sure that it actually is constant-time. If you're not doing that, or you're not verifying the generated assembly itself, your constant-time guarantee is not worth the paper it's printed on. Even if it's not even printed on any paper.
Speaking as a compiler user, how much longer until we can expect better support for Mixed Boolean Arithmetic (MBA) simplification from compilers? For example, it's problematic that GCC and Clang aren't able to tell that `((((x ^ y) | (~(x ^ y) + 1)) >> 63) - 1)` is equivalent to `x==y` or that `(((x ^ ((x ^ y) | ((x - y) ^ y))) >> 63) - 1)` is equivalent to `x>=y`. We need MBA simplification because the algebra engines don't support it and malware authors frequently use it to obfuscate programs. But if we could plug it into a C compiler that shows us in assembly what the code is actually doing, then it would help a lot with software analysis.
As such, if you're using a standard compiler for a language to implement a constant-time guarantee, you need to be doing verification of the resulting assembly to make sure that it actually is constant-time. If you're not doing that, or you're not verifying the generated assembly itself, your constant-time guarantee is not worth the paper it's printed on. Even if it's not even printed on any paper.