There are three parts to interrupt latency:
Normally, the operating system won't have any influence over 1. The operating system certainly influences 2. For example, an operating system will sometimes disable interrupts [to avoid an interrupt interfering with some critical operation, such as for example modifying something to do with interrupt handling, or when scheduling a new task, or even when executing in an interrupt handler. Some operating systems may disable interrupts for several milliseconds, where a good realtime OS will not have interrupts disabled for more than microseconds at the most.
And of course, the time it takes from the first instruction in the interrupt handler runs, until the actual driver code or similar is running can be quite a few instructions, and the operating system is responsible for all of them.
For real time behaviour, it's often the "worst case" that matters, where in non-real time OS's, the overall execution time is much more important, so if it's quicker to not enable interrupts for a few hundred instructions, because it saves several instructions of "enable interrupts, then disable interrupts", a Linux or Windows type OS may well choose to do so.