Tesla has historically been slow delivering parts. When my car got messed up two and a half years ago, they told me six weeks, and I went ballistic, making three phone calls and three emails in an afternoon. Those were emails written in all caps, and phone calls where I got loud and irritated, which I seldom do. An hour or so later I got a call from the Tesla Parts people who operated the Western U.S. and said I would have my parts "tomorrow", which I knew wouldn't happen, but they did get them to me mostly in the next two to five days.
It was obvious to me that their parts department was in development and unorganized. They are still working on it, know it's a problem, and want to do better. But... a couple months ago I tried to get parts for my daughter's X, and it still took 3 weeks, though this time no screaming or nasty emails. They just cannot do any better yet, though I know they want to.
By the way, you can save a lot of money by going to a NON-Tesla authorized repair facility. They get parts just as quickly, have the manuals, and the work I have seen is as good, only it's about half the price. And I support Tesla, just think they're overpriced.