complexity theory – Do time-constructible functions exist in relativized worlds?

I know that time-constructible functions are necessary to prove the Time Hierarchy Theorem and being computable functions they are computed by Turing Machines. I’m just confused in that since the Time Hierarchy Theorem relativizes I think this implies that TM’s computing time-constructible functions exist in relativized worlds as well. But since testing string membership in the oracle language happens instantly, I don’t see how they can be considered time-constructible. Is there another way to conceptualize it? Thanks.