If you were the first creative hire in a company with a legacy/outdated product that's never been touched by a designer, how long would you reasonably expect before being able to set in motion the implementation of a new design system and new UX?

This depends a lot on whether it is materially causing problems for the business. (There are an awful of legacy products out there with legacy UX, but there few motivations to improve them because cost of market entry is high, for example, and so there are few new competitors)

For the purposes of answering this question, I'm going to assume you are the first creative hire because there is a problem, and they want you to fix it. What I would probably not do is to go into meetings and pitch a new "design system" because that sounds a lot like airy-fairy academic noodling. Rather, pick one or two battles that you can win; investments you can make within a quarter or two that will drive material results. I'm suggesting, in other words, for you to flip the script: rather than "how long will it take to implement a new design system" it becomes "what can we do in two quarters?"

This doesn't mean you don't have a long range plan or a design system in mind upon which you are basing these investments. But use the near-term goal to keep you honest and grounded, as well as to pressure-test the design system. And the most critical investment here: find some way to measure the impact of design (changes) on the product and to the business goal you are focused on fixing. Good luck!